How to evaluate expressions in two variables with decimals and fractions

WEBVTT
Kind: captions
Language: en

00:00:00.258 --> 00:00:01.583
- [Voiceover] Let's see
if we can give ourselves

00:00:01.583 --> 00:00:04.252
some practice evaluating expressions

00:00:04.252 --> 00:00:06.552
that have two different variables in them

00:00:06.552 --> 00:00:08.270
So let's see if we can
evaluate the expression

00:00:08.270 --> 00:00:12.287
seven J plus five minus eight K,

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when J is equal to 0.5
and K is equal to 0.25.

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So why don't you try to pause the video

00:00:19.461 --> 00:00:22.702
and evaluate this first before
we work through it together.

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Alright, so if we want
to evaluate this thing,

00:00:26.167 --> 00:00:29.232
everywhere we see a J we
want to replace it with a 0.5

00:00:29.232 --> 00:00:30.556
and everywhere we see a K

00:00:30.556 --> 00:00:32.646
we want to replace it with a 0.25,

00:00:32.646 --> 00:00:33.412
so let's do that.

00:00:33.412 --> 00:00:35.873
This is going to be seven times,

00:00:35.873 --> 00:00:38.510
and instead of J I'm gonna put a zero,

00:00:39.444 --> 00:00:42.091
a 0.5 in there.

00:00:42.091 --> 00:00:45.667
And then we have plus five

00:00:45.667 --> 00:00:48.360
minus eight times K.

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And K we're saying is 0.25.

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0.25.

00:00:55.107 --> 00:00:56.797
So what is this going to be equal to?

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So if I were to take,

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if I were to take,

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and I can color code this,

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seven times 0.5.

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Half of 7, that's going to
be three and a half, 3.5.

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Then I have plus five.

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And then I'm gonna subtract.

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I am subtracting eight times 0.25.

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0.25, this is 1/4.

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I could rewrite this if I want.

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0.25, that's the same thing as 1/4.

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Eight times 1/4, or another
way to think about it is

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eight divided by four is
gonna be equal to two.

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So this whole thing over here
is going to be equal to two.

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So it's gonna be minus, we
have this minus out here,

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so minus two.

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And what is this going to be?

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Well, let's think about it.

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3.5 plus five is 8.5,

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minus two is going to be 6.5.

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So this is equal to this,

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is equal to 6.5.

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Let's do another one of these.

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Alright.

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And we'll, just like before,

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try to work through it on your own

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before we do it together.

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Alright, now let's do it together.

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So over here I have this expression 0.1 M

00:02:04.370 --> 00:02:07.550
plus eight minus 12 N,

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when M is equal to 30
and N is equal to 1/4.

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Alright.

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So everywhere I see an M I
want to replace with a 30.

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And everywhere I see an N I
want to replace with a 1/4.

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So this is going to be equal to

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0.1 times M.

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M is 30.

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Times 30 plus

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eight minus 12 times N,

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where N is 1/4.

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N is 1/4.

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So what is,

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what is

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1/10,

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This right over here,
0.1, that's the same thing

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as 1/10 of 30?

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Well 1/10 of 30, that's going to be three.

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So this part is three.

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And we have three plus eight.

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And then we're gonna have minus.

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Well what is 12 times 1/4?

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That's gonna be 12/4,
or 12 divided by four,

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which is going to be equal to three.

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And now when we evaluate this,

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so that is equal to this,

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we have three plus eight minus three.

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Well, threes are going to,

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you know positive three, then
you're gonna subtract three,

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and you're just going to be left with,

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you're just going to
be left with an eight.

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And you're done.

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This expression when M is equal
to 30 and N is equal to 1/4

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is equal to eight.


How to evaluate expressions with two variables

WEBVTT
Kind: captions
Language: en

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Now, let's think
about expressions

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with more than one variable.

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So say I had the
expression a plus--

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I'll do a really
simple one, a plus b.

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And I want to evaluate this
expression when a is equal to 7

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and b is equal to 2.

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And I encourage you to pause
this and try this on your own.

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Well, wherever we see
the a, we would just

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replace it with the 7.

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And wherever we see the b,
we'd replace it with the 2.

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So when a equals
7 and b equals 2,

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this expression
will be 7 plus 2,

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which, of course, is equal to 9.

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So this expression
would be equal to 9

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in this circumstance.

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Let's do a slightly
more complicated one.

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Let's say we have the expression
x times y minus y plus x.

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Actually, let's make it plus 3x.

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Or another way of saying
it plus 3 times x.

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So let's evaluate this when x is
equal to 3 and y is equal to 2.

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And once again, I encourage
you to pause this video

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and try this on your own.

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Well, everywhere we see an
x, let's replace it with a 3.

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Every place we see a y,
let's replace it with a 2.

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So this is going to
be equal to 3 times y.

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And y is 2 in this case.

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3 times 2 minus 2
plus this 3 times x.

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But x is also now equal to 3.

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So what is this
going to be equal to?

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Well, this is going to be
equal to 3 times 2 is 6.

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This 3 times 3 is 9.

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So it simplifies
to 6 minus 2, which

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is 4, plus 9, which
is equal to 13.

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So in this case,
it is equal to 13.


Coordinate plane: have all the points been graphed?

WEBVTT
Kind: captions
Language: en

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Consider the following
coordinate pairs.

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And they give us a
bunch right over here.

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Which of those pairs
is not graphed below?

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So let's see, 3
comma negative 2.

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So 3 comma negative 2,
well, actually, that

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looks like that first
one is not graphed.

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So 3 comma negative
2 isn't graphed.

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And you can verify that
all of these others

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do seem to be graphed.

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4 comma negative
8 is right there.

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Negative 8 comma 3
is right over here.

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4 comma 6 is right there.

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And negative 1 comma
0 is right there.

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So let's do a couple
more of these.

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So is negative 5
comma 0 graphed?

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Yep.

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It's right over there.

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Negative 4 comma positive
5 should be there.

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So that's not graphed.

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So let me just select that one.

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Let's do one more of these.

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Negative 8 comma 4, so negative
8 comma 4, we got lucky.

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The first one is not
graphed-- negative 8 comma 4.

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And we're done.


Order of operations examples: exponents

WEBVTT
Kind: captions
Language: en

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So I have six
different expressions

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here, and what I want you
to do is pause this video

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and try to calculate the value
of each of these expressions.

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I'm assuming you've
given a go at it.

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Now let's work through them.

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So when we see
something like this,

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we have to remember our
order of operations.

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We have 2 times
3 squared, and we

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have to remember that the
first thing we would need

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to think about are
the parentheses.

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I'll just write paren for short.

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Then we worry about exponents.

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Then we will worry about
multiplication and division,

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and actually let me
write it this way.

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We worry about
multiplication and division.

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And then we worry about
addition and subtraction.

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So in this expression
right over here,

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there are no parentheses, so
we do the exponents first.

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So we calculate
what 3 squared is.

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3 times 3 is 9, so
this becomes 2 times 9,

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which is equal to 18.

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Now let's look at this one,
and this one is interesting,

00:01:01.990 --> 00:01:04.879
because they have-- it looks
like the same expression,

00:01:04.879 --> 00:01:06.610
but now there are parentheses.

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And because of
these parentheses,

00:01:08.590 --> 00:01:10.090
we're going to do
the multiplication

00:01:10.090 --> 00:01:12.540
before we take the exponent.

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So 2 times 3 is going
to be 6, and we're

00:01:16.200 --> 00:01:19.440
going to take that
to the second power.

00:01:19.440 --> 00:01:25.350
So that's 6 times 6,
which is equal to 36.

00:01:25.350 --> 00:01:27.520
Now let's think about
this one right over here.

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Once again, we want to do our
multiplication and our division

00:01:30.630 --> 00:01:31.310
first.

00:01:31.310 --> 00:01:33.110
So we have a division
right over here.

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81/9 is the same thing
as 81 divided by 9,

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and that's going to be 9.

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And then we have-- so it
becomes 1 plus 5 times 9.

00:01:43.950 --> 00:01:46.140
Now we want to do the
multiplication before we

00:01:46.140 --> 00:01:49.050
do the addition, so
we're going to do our 5

00:01:49.050 --> 00:01:53.410
times 9, which is 45.

00:01:53.410 --> 00:01:59.940
So this becomes 1 plus 45,
which of course is equal to 46.

00:01:59.940 --> 00:02:03.020
Now let's tackle this
one right over here.

00:02:03.020 --> 00:02:06.150
So, we would want to
do the exponents first.

00:02:06.150 --> 00:02:09.230
So, 1 squared, well
that's just going--

00:02:09.230 --> 00:02:10.729
let me do this in
a different color.

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1 squared is just
going to be equal to 1,

00:02:13.930 --> 00:02:16.370
so that's just going
to be equal to 1.

00:02:16.370 --> 00:02:18.980
And so you have
2 times 4 plus 1.

00:02:18.980 --> 00:02:19.820
What should you do?

00:02:19.820 --> 00:02:21.986
Should you add first or do
the multiplication first?

00:02:21.986 --> 00:02:25.240
Well multiplication takes
precedence over addition,

00:02:25.240 --> 00:02:28.230
so you're going to do
the 2 times 4 first.

00:02:28.230 --> 00:02:30.950
2 times 4 is 8, so
you're going to have

00:02:30.950 --> 00:02:37.850
8 plus 1, which of
course is equal to 9.

00:02:37.850 --> 00:02:39.760
Now you have a very
similar expression,

00:02:39.760 --> 00:02:41.120
but you have parentheses.

00:02:41.120 --> 00:02:43.620
So that's going to force you
to do what's in the parentheses

00:02:43.620 --> 00:02:44.909
before you take the exponent.

00:02:44.909 --> 00:02:46.950
But within the parentheses
we have multiplication

00:02:46.950 --> 00:02:49.320
and addition, and
we have to remember

00:02:49.320 --> 00:02:51.990
that we do the
multiplication first.

00:02:51.990 --> 00:02:53.480
So we're going to
do the 2 times 4

00:02:53.480 --> 00:02:59.880
first, so that's going to be
8 plus 1 to the second power.

00:02:59.880 --> 00:03:03.370
8 plus 1 is 9, so that's
9 to the second power.

00:03:03.370 --> 00:03:09.671
9 squared is the same thing as
9 times 9, which is equal to 81.

00:03:09.671 --> 00:03:11.170
Now we have one
more right over here

00:03:11.170 --> 00:03:12.400
that looks very
similar to this one,

00:03:12.400 --> 00:03:14.350
except, once again, we
have parentheses that's

00:03:14.350 --> 00:03:15.855
making us do the addition first.

00:03:15.855 --> 00:03:17.980
Without parentheses, we
would do the multiplication

00:03:17.980 --> 00:03:19.850
and the division first.

00:03:19.850 --> 00:03:23.620
But here, we see
that 1 plus 5 is 6,

00:03:23.620 --> 00:03:27.400
and then we have this
81/9, which is 9.

00:03:27.400 --> 00:03:33.100
So this simplifies to 6 times 9,
which of course is equal to 54.


The zeroth power

WEBVTT
Kind: captions
Language: en

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- If we think about something
like two to the third power,

00:00:03.750 --> 00:00:06.210
we could view this as taking three twos

00:00:06.210 --> 00:00:10.570
and multiplying them together,
so two times two times two,

00:00:10.570 --> 00:00:13.090
or equivalently, we could
say this is the same thing

00:00:13.090 --> 00:00:17.460
as taking a one and then
multiplying it by two three times,

00:00:17.460 --> 00:00:19.120
so actually, let's just
go with this definition

00:00:19.120 --> 00:00:20.430
right over here, and this, of course,

00:00:20.430 --> 00:00:23.790
is going to be equal to eight.

00:00:23.790 --> 00:00:27.050
Now what would, based on
this definition I just did,

00:00:27.050 --> 00:00:29.720
what would two to the second power be?

00:00:29.720 --> 00:00:34.090
Well, this would be one times two twice,

00:00:34.090 --> 00:00:36.790
so one times two times two, which,

00:00:36.790 --> 00:00:39.330
of course, would be equal to four.

00:00:39.330 --> 00:00:42.820
What would two to the first power be?

00:00:42.820 --> 00:00:44.300
Well, that would be one, and we would

00:00:44.300 --> 00:00:48.920
multiply it by one two, one times two,

00:00:48.920 --> 00:00:51.230
which, of course, is equal to two.

00:00:51.230 --> 00:00:54.020
Now let's ask ourself
an interesting question.

00:00:54.020 --> 00:00:57.420
Based on this definition
of what an exponent is,

00:00:57.420 --> 00:01:01.530
what would two to the zeroth power be?

00:01:01.530 --> 00:01:03.720
I encourage you to just think
about that a little bit.

00:01:03.720 --> 00:01:05.690
If you were the mathematics community,

00:01:05.690 --> 00:01:07.890
how would you define two to the zero power

00:01:07.890 --> 00:01:11.390
so it is consistent with
everything that we just saw.

00:01:11.390 --> 00:01:14.930
Well, the way we just talked
about it, we just said

00:01:14.930 --> 00:01:17.800
exponentiation is you start
with a one and you multiply it

00:01:17.800 --> 00:01:20.990
by the base zero times, so
we're not gonna multiply it

00:01:20.990 --> 00:01:24.330
by any two, so we're just
gonna be left with a one.

00:01:24.330 --> 00:01:26.130
So does this make sense that two

00:01:26.130 --> 00:01:29.390
to the zero power is equal to one?

00:01:29.390 --> 00:01:31.220
Well, let's think about it another way,

00:01:31.220 --> 00:01:33.820
and let's do a different base.

00:01:33.820 --> 00:01:38.820
That was with two, but
let's say we have three

00:01:39.610 --> 00:01:42.190
and I could say three to
the fourth power, that's

00:01:42.190 --> 00:01:46.550
three times three times three times three

00:01:46.550 --> 00:01:48.979
which is going to be equal to 81,

00:01:48.979 --> 00:01:50.887
and let me just write down that

00:01:50.887 --> 00:01:55.030
this is going to be equal to 81.

00:01:55.030 --> 00:01:57.470
If I said three to the third power,

00:01:57.470 --> 00:02:01.790
that's three times three
times three which is 27.

00:02:01.790 --> 00:02:05.460
Three to the second
power is equal to nine.

00:02:05.460 --> 00:02:08.150
Three to the first
power is equal to three.

00:02:08.150 --> 00:02:10.360
Do you notice a pattern every time we

00:02:10.360 --> 00:02:12.390
decrease the exponent here by one?

00:02:12.390 --> 00:02:13.690
We want three to the fourth, and now we

00:02:13.690 --> 00:02:14.550
go three to the third.

00:02:14.550 --> 00:02:17.450
What happened to the value?

00:02:17.450 --> 00:02:22.114
Well, going from 81 to 27,
we divided by three, and

00:02:22.114 --> 00:02:23.630
that makes sense,
because we're multiplying

00:02:23.630 --> 00:02:26.090
by one less three, so we divide by three

00:02:26.090 --> 00:02:29.900
to go from 81 to 27, we
divide by three again

00:02:29.900 --> 00:02:32.522
if our exponent goes down by one,

00:02:32.522 --> 00:02:36.300
and we divide by three again
when we go from nine to three,

00:02:36.300 --> 00:02:39.500
divide by three, so based
on this, what do you think

00:02:39.500 --> 00:02:43.350
three to the zero power should be?

00:02:43.350 --> 00:02:45.250
Well, the pattern is every time we

00:02:45.250 --> 00:02:49.290
decrease our exponent by
one, we divide by the base,

00:02:49.290 --> 00:02:51.570
and so we should divide by three again

00:02:51.570 --> 00:02:54.120
would be the logic if
we follow that pattern,

00:02:54.120 --> 00:02:58.120
and so three divided by
three would get us one again.

00:02:58.120 --> 00:03:01.736
So I know it might seem a little
bit counter intuitive that

00:03:01.736 --> 00:03:05.760
something to the zeroth power
is going to be equal to one,

00:03:05.760 --> 00:03:08.650
but this is how the mathematics
community has defined it

00:03:08.650 --> 00:03:10.930
because it actually makes a lot of sense.

00:03:10.930 --> 00:03:13.720
Either if you view an
exponent as taking a one

00:03:13.720 --> 00:03:17.170
and multiplying it by the base
the exponent number of times,

00:03:17.170 --> 00:03:19.850
so I'm gonna multiply
one by two three times,

00:03:19.850 --> 00:03:23.420
or if you just follow
this pattern, every time

00:03:23.420 --> 00:03:25.720
you decrease the exponent by one, you're

00:03:25.720 --> 00:03:27.960
going to be dividing by the base.

00:03:27.960 --> 00:03:31.050
Either of those would get
you to the conclusion that

00:03:31.050 --> 00:03:33.860
two to the zero power is one,
or three to the zero power

00:03:33.860 --> 00:03:38.090
is one, or frankly, any number
to the zero power is one.

00:03:38.090 --> 00:03:43.090
So if I have any number,
let's say I have some number a

00:03:43.110 --> 00:03:46.950
to the zero power, this is
going to be equal to one.

00:03:46.950 --> 00:03:48.930
Now I have an interesting
question for you,

00:03:48.930 --> 00:03:50.584
and let's just say this is the case

00:03:50.584 --> 00:03:53.470
when a does not equal zero.

00:03:53.470 --> 00:03:54.850
I'll leave you a little bit of a

00:03:54.850 --> 00:03:56.590
puzzle for you to think about.

00:03:56.590 --> 00:04:00.290
What do you think is
zero to the zeroth power?

00:04:00.290 --> 00:04:04.090
What should zero to the zero power be?

00:04:04.090 --> 00:04:08.460
And what's interesting
about zero to the zero power

00:04:08.460 --> 00:04:11.350
is you'll get a different
answer if you use this technique

00:04:11.350 --> 00:04:15.890
versus if you use this
technique right over here.

00:04:15.890 --> 00:04:18.882
This technique would actually
get you to being one,

00:04:18.882 --> 00:04:21.580
while this technique would
have you divide by zero,

00:04:21.580 --> 00:04:23.160
which we don't know how to do.

00:04:23.160 --> 00:04:25.420
Anyway, I'll leave you there to ponder

00:04:25.420 --> 00:04:30.370
the mysteries of zero to the zeroth power.


Dividing decimals with hundredths

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:06.530
Let's see if we can divide
30.24 divided by 0.42.

00:00:06.530 --> 00:00:11.160
And try pausing the video
and solving it on your own

00:00:11.160 --> 00:00:12.654
before I work through it.

00:00:12.654 --> 00:00:14.820
So there is a couple of
ways you can think about it.

00:00:14.820 --> 00:00:26.590
We could just write it
as 30.24 divided by 0.42.

00:00:26.590 --> 00:00:28.240
But what do you do now?

00:00:28.240 --> 00:00:30.110
Well the important
realization is,

00:00:30.110 --> 00:00:32.750
is when you're doing a
division problem like this,

00:00:32.750 --> 00:00:35.260
you will get the
same answer as long

00:00:35.260 --> 00:00:38.880
as you multiply or divide both
numbers by the same thing.

00:00:38.880 --> 00:00:42.850
And to understand that,
rewrite this division

00:00:42.850 --> 00:00:50.200
as 30.42 over 0.42.

00:00:50.200 --> 00:00:53.620
We could write it
really as a fraction.

00:00:53.620 --> 00:00:55.920
And we know that when we
have a fraction like this

00:00:55.920 --> 00:00:57.950
we're not changing the
value of the fraction

00:00:57.950 --> 00:01:00.110
if we multiple the numerator
and the denominator

00:01:00.110 --> 00:01:01.480
by the same quantity.

00:01:01.480 --> 00:01:04.010
And so what could we
multiply this denominator by

00:01:04.010 --> 00:01:05.610
to make it a whole number?

00:01:05.610 --> 00:01:09.460
Well we can multiply it
by 10 and then another 10.

00:01:09.460 --> 00:01:11.070
So we can multiply it by a 100.

00:01:11.070 --> 00:01:11.950
So lets do that.

00:01:11.950 --> 00:01:14.650
If we multiply the
denominator by 100 in order

00:01:14.650 --> 00:01:17.100
to not change the
value of this, we also

00:01:17.100 --> 00:01:19.500
need to multiply the
numerator by 100.

00:01:19.500 --> 00:01:22.140
We are essentially
multiplying by 100 over 100,

00:01:22.140 --> 00:01:23.030
which is just 1.

00:01:23.030 --> 00:01:25.960
So we're not changing the
value of this fraction.

00:01:25.960 --> 00:01:30.330
Or, you could view this,
this division problem.

00:01:30.330 --> 00:01:34.020
So this is going to
be 30.42 times 100.

00:01:34.020 --> 00:01:39.020
Move the decimal two places
to the right, gets you 3,042.

00:01:39.020 --> 00:01:41.430
The decimal is now there
if you care about it.

00:01:41.430 --> 00:01:44.460
And, 0.42 times 100.

00:01:44.460 --> 00:01:47.820
Once again move the decimal
one, two places to the right,

00:01:47.820 --> 00:01:50.400
it is now 42.

00:01:50.400 --> 00:01:54.360
So this is going to be the
exact same thing as 3,042

00:01:54.360 --> 00:01:56.920
divided by 42.

00:01:56.920 --> 00:02:01.379
So once again we can move the
decimal here, two to the right.

00:02:01.379 --> 00:02:02.920
And if we move that
two to the right,

00:02:02.920 --> 00:02:05.330
then we can move this
two to the right.

00:02:05.330 --> 00:02:07.170
Or we need to move
this two to the right.

00:02:07.170 --> 00:02:10.130
And so this is where,
now the decimal place is.

00:02:10.130 --> 00:02:14.260
You could view this as 3,024.

00:02:14.260 --> 00:02:18.950
Let me clear that
3024 divided by 42.

00:02:18.950 --> 00:02:20.950
Let me clear that.

00:02:20.950 --> 00:02:22.950
And we know how to tackle
that already, but lets

00:02:22.950 --> 00:02:24.210
do it step by step.

00:02:24.210 --> 00:02:26.940
How many times
does 42 go into 3?

00:02:26.940 --> 00:02:29.670
Well it does not go at all,
so we can move on to 30.

00:02:29.670 --> 00:02:31.380
How many times
does 42 go into 30?

00:02:31.380 --> 00:02:35.060
Well it does not go into 30
so we can move on to 302.

00:02:35.060 --> 00:02:38.320
How many times does
42 go into 302?

00:02:38.320 --> 00:02:41.250
And like always this is a bit
of an art when your dividing

00:02:41.250 --> 00:02:44.230
by a two-digit or a multi-digit
number, I should say.

00:02:44.230 --> 00:02:46.110
So lets think about
it a little bit.

00:02:46.110 --> 00:02:47.720
So this is roughly 40.

00:02:47.720 --> 00:02:49.770
This is roughly 300.

00:02:49.770 --> 00:02:52.390
So how many times
does 40 go into 300?

00:02:52.390 --> 00:02:55.980
Well how many times
does 4 go into 30?

00:02:55.980 --> 00:02:57.810
Well, it looks like
it's about seven times,

00:02:57.810 --> 00:03:00.320
so I'm going to try out
a 7, see if it works out.

00:03:00.320 --> 00:03:03.630
7 times 2 is 14.

00:03:03.630 --> 00:03:06.250
7 times 4 is 28.

00:03:06.250 --> 00:03:08.590
Plus 1 is 29.

00:03:08.590 --> 00:03:10.430
And now I can subtract.

00:03:10.430 --> 00:03:12.570
Do a little bit of
re-grouping here.

00:03:12.570 --> 00:03:17.320
So lets see, if I regroup--
I take a 100 from the 300.

00:03:17.320 --> 00:03:18.730
That becomes a 200.

00:03:18.730 --> 00:03:22.250
Then our zero tens,
now I have 10 tens,

00:03:22.250 --> 00:03:24.220
but I'm going to need
one of those 10 tens,

00:03:24.220 --> 00:03:25.430
so that's going to be 9 tens.

00:03:25.430 --> 00:03:26.888
And I'm going to
give it over here.

00:03:26.888 --> 00:03:28.780
So this is going to be a 12.

00:03:28.780 --> 00:03:34.670
12 minus 4 is 8.

00:03:34.670 --> 00:03:36.780
9 minus 9 is 0.

00:03:36.780 --> 00:03:39.490
2 minus 2 is 0.

00:03:39.490 --> 00:03:42.620
So what I got left
over is less than 42,

00:03:42.620 --> 00:03:44.447
so I know that 7 is
the right number.

00:03:44.447 --> 00:03:46.030
I want to go as many
times as possible

00:03:46.030 --> 00:03:48.420
into 302 without going over.

00:03:48.420 --> 00:03:51.840
So now lets bring
down the next digit.

00:03:51.840 --> 00:03:53.990
Lets bring down
this 4 over here.

00:03:53.990 --> 00:03:56.669
How many times
does 42 go into 84?

00:03:56.669 --> 00:03:58.460
Well that jumps out at
you, or hopefully it

00:03:58.460 --> 00:04:02.470
jumps-- It goes two times.

00:04:02.470 --> 00:04:04.750
2 times 2 is 4.

00:04:04.750 --> 00:04:07.840
2 times 4 is 8.

00:04:07.840 --> 00:04:11.030
You subtract, and we
have no remainder.

00:04:11.030 --> 00:04:16.120
So 3,042 divided by
42 is the same thing

00:04:16.120 --> 00:04:19.060
as 30.42 divided by 0.42.

00:04:19.060 --> 00:04:21.752
And it's going to
be equal to 72.

00:04:25.477 --> 00:04:27.185
Actually, I didn't
have to copy and paste

00:04:27.185 --> 00:04:28.310
that, I'll just write this.

00:04:28.310 --> 00:04:29.480
This is equal to 72.

00:04:34.370 --> 00:04:36.450
Just like that.


Dividing by two digits example 1

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.470
Let's divide 9,815
by 65, or figure out

00:00:04.470 --> 00:00:07.950
how many times the
65 go into 9,815.

00:00:07.950 --> 00:00:09.640
And I encourage you
to pause this video

00:00:09.640 --> 00:00:11.950
and try this on your own.

00:00:11.950 --> 00:00:21.557
So let me just rewrite this
as 9,815 divided by 65.

00:00:21.557 --> 00:00:23.390
And we write it this
way because it's easier

00:00:23.390 --> 00:00:25.200
to manipulate the
numbers, kind of doing

00:00:25.200 --> 00:00:27.170
the standard process here.

00:00:27.170 --> 00:00:29.180
And as we'll see,
whenever we divide

00:00:29.180 --> 00:00:31.289
by a number that has
more than one digit,

00:00:31.289 --> 00:00:32.830
there's a little
bit of an art to it.

00:00:32.830 --> 00:00:34.660
And hopefully you'll
get an appreciation

00:00:34.660 --> 00:00:36.822
for that art over the
course of this video.

00:00:36.822 --> 00:00:38.280
So first we could
think about well,

00:00:38.280 --> 00:00:41.577
how many times
does 65 go into 9?

00:00:41.577 --> 00:00:43.160
Well it doesn't go
into 9 at all so we

00:00:43.160 --> 00:00:44.760
can move one digit to the right.

00:00:44.760 --> 00:00:49.200
How many times does it go
into 98 without going over it?

00:00:49.200 --> 00:00:53.170
Well 65 times 1 is 65 so
that doesn't go over it.

00:00:53.170 --> 00:00:55.920
And 65 times 2, well
that would be 130 so

00:00:55.920 --> 00:00:57.540
that would go over 98.

00:00:57.540 --> 00:01:00.520
So it only goes one time.

00:01:00.520 --> 00:01:04.819
We multiply 1 times
65, which is 65.

00:01:04.819 --> 00:01:08.210
And then we could subtract to
see how much we have left over.

00:01:08.210 --> 00:01:13.760
So 8 minus 5 is 3
and 9 minus 6 is 3.

00:01:13.760 --> 00:01:18.477
And now we can bring down
the next digit, this 1 here.

00:01:18.477 --> 00:01:20.060
And now this is where
the art is going

00:01:20.060 --> 00:01:22.018
to come into play because
we need to figure out

00:01:22.018 --> 00:01:28.685
how many times does 65 go into
331 without going over it.

00:01:28.685 --> 00:01:31.020
And you might just try
to look at these numbers,

00:01:31.020 --> 00:01:33.030
try to approximate
them a little bit.

00:01:33.030 --> 00:01:36.950
You might say, well, maybe 65,
let me round this thing up.

00:01:36.950 --> 00:01:38.730
Maybe this is close to 70.

00:01:38.730 --> 00:01:43.170
And let's see, this
is close to 300.

00:01:43.170 --> 00:01:46.560
So maybe we say, well,
70 would go into 300.

00:01:46.560 --> 00:01:50.760
So maybe we think about how
many times does 70 go into 300?

00:01:50.760 --> 00:01:52.330
And we say without
going over it,

00:01:52.330 --> 00:01:53.980
it doesn't go exactly into 300.

00:01:53.980 --> 00:01:57.400
Well you could say, well how
many times does 7 go into 30?

00:01:57.400 --> 00:02:01.640
Well we know 7 goes
into 30 four times.

00:02:01.640 --> 00:02:03.520
4 times 7 is 28.

00:02:03.520 --> 00:02:06.010
So maybe try a 4 right
over here because then this

00:02:06.010 --> 00:02:08.777
will be 280, 4 times 70 is 280.

00:02:08.777 --> 00:02:10.860
You're still going to have
a little bit left over,

00:02:10.860 --> 00:02:13.890
but what you have left over
is going to be less than 70.

00:02:13.890 --> 00:02:15.520
It's going to be 20.

00:02:15.520 --> 00:02:17.680
So you say, well, if
this is roughly 70

00:02:17.680 --> 00:02:20.090
and if this is roughly
300, then maybe it's

00:02:20.090 --> 00:02:21.410
going to be the same thing.

00:02:21.410 --> 00:02:22.750
So let's try that out.

00:02:22.750 --> 00:02:25.590
Let's see if it goes four times.

00:02:25.590 --> 00:02:31.440
So 4 times 5 is 20, carry the 2.

00:02:31.440 --> 00:02:35.810
4 times 6 is 24 plus 2 is 26.

00:02:35.810 --> 00:02:38.070
And now let's see how
much we had left over.

00:02:38.070 --> 00:02:42.663
So when we subtract,
we are left with-- I'll

00:02:42.663 --> 00:02:46.260
do this in a new
color-- 1 minus 0 is 1.

00:02:46.260 --> 00:02:47.980
We have a 3 here and
a 6 here so we're

00:02:47.980 --> 00:02:49.080
going to have to do
a little regrouping.

00:02:49.080 --> 00:02:50.960
Let's take 100 from
the hundreds place.

00:02:50.960 --> 00:02:52.290
It becomes 200.

00:02:52.290 --> 00:02:56.170
Give those 10 tens, that
100, to the tens place.

00:02:56.170 --> 00:02:58.030
So now we have 13 tens.

00:02:58.030 --> 00:03:02.540
13 minus 6 is 7 and
then 2 minus 2 is 1.

00:03:02.540 --> 00:03:04.410
So did this work out?

00:03:04.410 --> 00:03:07.270
Well no, our remainder, after
we said it went in four times,

00:03:07.270 --> 00:03:09.480
we actually had 71 left over.

00:03:09.480 --> 00:03:13.717
71, this right over
here, is larger than 65.

00:03:13.717 --> 00:03:16.050
You don't want a situation
where what you have left over

00:03:16.050 --> 00:03:17.425
is larger than
what you're trying

00:03:17.425 --> 00:03:18.630
to divide into the number.

00:03:18.630 --> 00:03:20.180
You could have gone
into it one more

00:03:20.180 --> 00:03:23.980
time because you had
so much left over.

00:03:23.980 --> 00:03:26.270
So this 4 was actually too low.

00:03:26.270 --> 00:03:28.900
We should have probably
approximated this as 60,

00:03:28.900 --> 00:03:31.930
and 60 goes into 300,
if we were to estimate,

00:03:31.930 --> 00:03:34.360
we'd say, well that might
be closer to five times.

00:03:34.360 --> 00:03:37.230
So this is where the art
of this comes into play.

00:03:37.230 --> 00:03:40.600
So it was very reasonable to
do what I just did, but it just

00:03:40.600 --> 00:03:43.680
turned out to not be the
right way to think about it.

00:03:43.680 --> 00:03:45.640
I could just say, well
the 4 wasn't enough.

00:03:45.640 --> 00:03:46.740
I had too much left over.

00:03:46.740 --> 00:03:49.050
Let me try 5 now.

00:03:49.050 --> 00:03:53.110
5 times 5 is 25, carry the 2.

00:03:53.110 --> 00:03:56.360
5 times 6 is 30, plus 2 is 32.

00:03:56.360 --> 00:03:56.930
There you go.

00:03:56.930 --> 00:04:00.390
We got much closer to
331 without going over.

00:04:00.390 --> 00:04:04.010
Now we can subtract.

00:04:04.010 --> 00:04:06.070
And once again, we could
do a little regrouping.

00:04:06.070 --> 00:04:07.860
Take a 10 from the tens place.

00:04:07.860 --> 00:04:09.240
This becomes two tens.

00:04:09.240 --> 00:04:11.270
This becomes an 11.

00:04:11.270 --> 00:04:15.460
11 minus 5 is 6, 2 minus
2 is 0, 3 minus 3 is 0.

00:04:15.460 --> 00:04:17.440
So we only have 6
left over, which

00:04:17.440 --> 00:04:19.370
is obviously less than 65.

00:04:19.370 --> 00:04:20.640
So we're all good.

00:04:20.640 --> 00:04:24.070
And if we put a 6 here, we
would have gone over 331.

00:04:24.070 --> 00:04:26.460
And so that wouldn't
have been cool either.

00:04:26.460 --> 00:04:29.480
But anyway, let's bring
down the next digit.

00:04:29.480 --> 00:04:31.260
Let's bring down the 5.

00:04:31.260 --> 00:04:33.826
So how many times
does 65 go into 65?

00:04:33.826 --> 00:04:35.530
Well, it goes one time.

00:04:35.530 --> 00:04:36.920
1 times 65,-- OK.

00:04:36.920 --> 00:04:39.220
Ignore this, that's from
a previous step-- 1 times

00:04:39.220 --> 00:04:41.620
65 is 65.

00:04:41.620 --> 00:04:46.740
And then you subtract,
and we have no remainder.

00:04:46.740 --> 00:04:57.272
So we see that 65 goes into
9,815 exactly 150-- let me

00:04:57.272 --> 00:05:00.650
just that in that
same blue color,

00:05:00.650 --> 00:05:08.800
I don't want to do all these
arbitrary colors-- 151 times.


Subtracting decimals example 2

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:07.090
Let's try to calculate
39.1 minus 0.794,

00:00:07.090 --> 00:00:11.210
and so pause the video
and try this on your own.

00:00:11.210 --> 00:00:13.649
All right, I'm assuming you've
given a go at it, so now

00:00:13.649 --> 00:00:14.940
let's work through it together.

00:00:14.940 --> 00:00:16.148
So I'm going to rewrite this.

00:00:16.148 --> 00:00:21.660
It's 39.1 minus-- I'm going to
line up the decimals so that I

00:00:21.660 --> 00:00:24.110
have the right place values
below the right place

00:00:24.110 --> 00:00:27.240
values-- minus-- this
0 is in the ones place,

00:00:27.240 --> 00:00:33.790
so I'll put it in the
ones place-- 0.794.

00:00:33.790 --> 00:00:37.130
And now we're ready to subtract.

00:00:37.130 --> 00:00:39.770
Now, how do we subtract
4 from nothingness here,

00:00:39.770 --> 00:00:41.030
and 9 from nothingness here?

00:00:41.030 --> 00:00:45.670
Well, the same thing
as nothing is a 0.

00:00:45.670 --> 00:00:48.797
And so now we can start to
think about how to subtract.

00:00:48.797 --> 00:00:50.130
Well, we still have the problem.

00:00:50.130 --> 00:00:52.070
Well, we're trying
to subtract 4 from 0,

00:00:52.070 --> 00:00:54.660
so we're trying to
subtract 9 from 0.

00:00:54.660 --> 00:00:57.580
So what we could do
is take this one tenth

00:00:57.580 --> 00:01:02.000
and try to regroup it
into the hundredths place

00:01:02.000 --> 00:01:03.730
and the thousandths place.

00:01:03.730 --> 00:01:05.600
So let's think about this.

00:01:05.600 --> 00:01:08.576
If we make this-- actually
that's not actually

00:01:08.576 --> 00:01:09.700
going to solve our problem.

00:01:09.700 --> 00:01:11.100
Well we could do
it, but then we're

00:01:11.100 --> 00:01:12.849
going to have zero
tenths, and we're still

00:01:12.849 --> 00:01:14.120
going to have a problem here.

00:01:14.120 --> 00:01:15.840
So actually let me
go to the ones place.

00:01:15.840 --> 00:01:17.890
So let me get rid
of a ones, so that's

00:01:17.890 --> 00:01:20.919
eight ones, which is
going to be 10 tenths.

00:01:20.919 --> 00:01:23.210
So that's going to now--
we're going to have 11 tenths.

00:01:23.210 --> 00:01:26.390
The 10 tenths from here
plus 1 is 11 tenths.

00:01:26.390 --> 00:01:28.090
Now let's take one
of those tenths

00:01:28.090 --> 00:01:31.890
so that we have 10 tenths,
and give it to the hundredths.

00:01:31.890 --> 00:01:34.120
So that's going to
be 10 hundredths.

00:01:34.120 --> 00:01:36.050
And now let's take one
of those hundredths--

00:01:36.050 --> 00:01:39.120
so now we have nine hundredths--
and give it to the thousandths.

00:01:39.120 --> 00:01:41.650
So that's going to
be 10 thousandths.

00:01:41.650 --> 00:01:43.830
Now we're ready to subtract.

00:01:43.830 --> 00:01:49.180
So 10-- let me do this in
yellow-- 10 minus 4 is 6.

00:01:49.180 --> 00:01:51.610
9 minus 9 is 0.

00:01:51.610 --> 00:01:54.067
10 minus 7 is 3.

00:01:54.067 --> 00:01:55.150
We have our decimal point.

00:01:55.150 --> 00:01:58.410
8 minus 0 is 8.

00:01:58.410 --> 00:02:00.990
And then we have 3
minus nothing is 3.

00:02:00.990 --> 00:02:03.080
So we're done, 38.306.


Comparing distributions with dot plots (example problem)

WEBVTT
Kind: captions
Language: en

00:00:01.140 --> 00:00:02.760
At the Olympic
games, many events

00:00:02.760 --> 00:00:04.790
have several rounds
of competition.

00:00:04.790 --> 00:00:07.750
One of these is the men's
100 meter backstroke.

00:00:07.750 --> 00:00:09.710
The upper dot plot
shows the times

00:00:09.710 --> 00:00:11.620
in seconds of the
top 8 finishers

00:00:11.620 --> 00:00:14.980
in the semifinal round
at the 2012 Olympics.

00:00:14.980 --> 00:00:18.750
The lower dot plot shows the
times of the same 8 swimmers,

00:00:18.750 --> 00:00:21.230
but in the final round.

00:00:21.230 --> 00:00:22.980
Which pieces of
information can be

00:00:22.980 --> 00:00:24.430
gathered from these dot plots?

00:00:24.430 --> 00:00:27.450
In the semifinal round, we
see that these are the 8 times

00:00:27.450 --> 00:00:28.360
of the 8 swimmers.

00:00:28.360 --> 00:00:32.340
We see 3 swimmers finished
in exactly 53.5 seconds.

00:00:32.340 --> 00:00:38.480
One swimmer finished in
53.7 seconds right here.

00:00:38.480 --> 00:00:44.740
And one swimmer right over
here finished in 52.7 seconds.

00:00:44.740 --> 00:00:47.610
And we can think about similar
things for each of these dots.

00:00:47.610 --> 00:00:49.840
Now, in the final
round, one swimmer

00:00:49.840 --> 00:00:51.730
here went much,
much, much faster.

00:00:51.730 --> 00:00:55.170
So this is in 52.2 seconds.

00:00:55.170 --> 00:00:58.010
While this swimmer right
over here went slower.

00:00:58.010 --> 00:01:00.150
We don't know which
dot he was up here.

00:01:00.150 --> 00:01:02.730
But regardless of which
dot he was up here,

00:01:02.730 --> 00:01:05.740
this dot took more time
than all of these dots.

00:01:05.740 --> 00:01:07.490
So his time
definitely got worse.

00:01:07.490 --> 00:01:13.390
And this is at 53.8 seconds.

00:01:13.390 --> 00:01:16.930
So let's look at the statements
and see which of these apply.

00:01:16.930 --> 00:01:22.720
The swimmers had faster times,
on average, in the finals.

00:01:22.720 --> 00:01:23.490
Is this true?

00:01:23.490 --> 00:01:26.520
Faster times on
average in the finals?

00:01:26.520 --> 00:01:28.700
So if we look at the
finals right over here,

00:01:28.700 --> 00:01:30.780
we could take each
of these times,

00:01:30.780 --> 00:01:33.760
add them up, and then divide by
8, the number of times we have.

00:01:33.760 --> 00:01:36.330
But let's see if we can get an
intuition for where this is,

00:01:36.330 --> 00:01:40.220
because we're really
just comparing these two

00:01:40.220 --> 00:01:42.940
plots, or these two
distributions, we could say.

00:01:42.940 --> 00:01:46.350
And so let's see, if all
the data was these three

00:01:46.350 --> 00:01:48.200
points and these
three points, we

00:01:48.200 --> 00:01:52.840
could intuit that the mean
would be right around there.

00:01:52.840 --> 00:01:58.040
It would be around 53.2 or 53.3
seconds, right around there.

00:01:58.040 --> 00:02:00.394
And then we have this
point and this point,

00:02:00.394 --> 00:02:02.060
if you just found the
mean of that point

00:02:02.060 --> 00:02:04.190
and that point, so
halfway between that point

00:02:04.190 --> 00:02:06.290
and that point, would get
you right around there.

00:02:06.290 --> 00:02:08.889
So the mean of those two
points would bring down

00:02:08.889 --> 00:02:10.440
the mean a little bit.

00:02:10.440 --> 00:02:13.080
And once again, I'm not
figuring out the exact number.

00:02:13.080 --> 00:02:18.830
But maybe it would be around
53.2, 53.1, or 53.2 seconds.

00:02:18.830 --> 00:02:23.540
So that's my intuition for
the mean of the final round.

00:02:23.540 --> 00:02:28.560
And now let's think about the
mean of the semifinal round.

00:02:28.560 --> 00:02:31.286
Let's just look at
these bottom five dots.

00:02:31.286 --> 00:02:32.660
If you find their
mean, you could

00:02:32.660 --> 00:02:35.750
intuit it would be maybe
someplace around here,

00:02:35.750 --> 00:02:41.157
pretty close to
around 53.3 seconds.

00:02:41.157 --> 00:02:42.740
And then you have
all these other ones

00:02:42.740 --> 00:02:45.100
that are at 53.5
and 53.3, which will

00:02:45.100 --> 00:02:47.010
bring the mean even higher.

00:02:47.010 --> 00:02:49.660
So I think it's fair to say that
the mean in the final around

00:02:49.660 --> 00:02:52.170
and the time is less
than the mean up here.

00:02:52.170 --> 00:02:53.670
And you could
calculate it yourself,

00:02:53.670 --> 00:02:55.260
but I'm just trying to
look at the distributions

00:02:55.260 --> 00:02:56.600
and get an intuition here.

00:02:56.600 --> 00:02:58.810
And at least in this case,
it looks pretty clear

00:02:58.810 --> 00:03:01.930
that the swimmers had
faster times, on average,

00:03:01.930 --> 00:03:02.610
in the finals.

00:03:02.610 --> 00:03:04.740
It took them less time.

00:03:04.740 --> 00:03:07.524
One of the swimmers was
disqualified from the finals.

00:03:07.524 --> 00:03:08.440
Well, that's not true.

00:03:08.440 --> 00:03:10.420
We have 8 swimmers in
the semifinal round.

00:03:10.420 --> 00:03:12.240
And we have 8 swimmers
in the final round.

00:03:12.240 --> 00:03:13.930
So that one's not true.

00:03:13.930 --> 00:03:17.490
The times in the
finals vary noticeably

00:03:17.490 --> 00:03:20.250
more than the times
in the semifinals.

00:03:20.250 --> 00:03:21.880
That does look to be true.

00:03:21.880 --> 00:03:24.100
We see in the semifinals,
a lot of the times

00:03:24.100 --> 00:03:25.530
were clumped up
right around here

00:03:25.530 --> 00:03:29.600
at 53.3 seconds
and 53.5 seconds.

00:03:29.600 --> 00:03:32.660
The high time isn't
as high as this time.

00:03:32.660 --> 00:03:34.460
The low time isn't as low there.

00:03:34.460 --> 00:03:37.260
So the final round
is definitely--

00:03:37.260 --> 00:03:39.760
they vary noticeably more.

00:03:39.760 --> 00:03:43.087
Individually, the swimmers
all swam faster in the finals

00:03:43.087 --> 00:03:44.420
than they did in the semifinals.

00:03:44.420 --> 00:03:45.730
Well, that's not true.

00:03:45.730 --> 00:03:48.530
Whoever this was, clearly they
were one of these data points

00:03:48.530 --> 00:03:49.030
up here.

00:03:49.030 --> 00:03:51.860
This data point took more time
than all of these data points.

00:03:51.860 --> 00:03:54.820
So this represents someone who
took more time in the finals

00:03:54.820 --> 00:03:57.102
than they did in the semifinals.

00:03:57.102 --> 00:03:59.540
And we got it right.


Count outcomes using tree diagram

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.390
Let's say that I've just
won some type of contest

00:00:03.390 --> 00:00:05.210
at a car dealership,
and they're going

00:00:05.210 --> 00:00:08.029
to give me a brand new car.

00:00:08.029 --> 00:00:09.900
And in deciding which
car they give me,

00:00:09.900 --> 00:00:12.850
they're first going to randomly
select the engine type.

00:00:12.850 --> 00:00:16.740
So the engine will come in
two different varieties.

00:00:16.740 --> 00:00:23.000
It'll either be a four-cylinder
or a six-cylinder engine.

00:00:23.000 --> 00:00:25.370
And they're literally just
going to flip a fair coin

00:00:25.370 --> 00:00:27.720
to decide whether I get
a four-cylinder engine

00:00:27.720 --> 00:00:30.280
or a six-cylinder engine.

00:00:30.280 --> 00:00:32.070
Then they're going
to pick the color.

00:00:32.070 --> 00:00:35.800
And there's four different
colors that the cars come in.

00:00:35.800 --> 00:00:38.340
So I'll write color
in a neutral color.

00:00:38.340 --> 00:00:40.800
So you could get a red car.

00:00:40.800 --> 00:00:42.070
That's not red.

00:00:42.070 --> 00:00:44.910
Let me do that in actual
red color, or close to red.

00:00:44.910 --> 00:00:49.580
You could get a red car,
you could get a blue car,

00:00:49.580 --> 00:00:58.730
you could get a green car,
or you could get a white car.

00:00:58.730 --> 00:01:03.532
And once again, they're going
to have the red, blue, green,

00:01:03.532 --> 00:01:05.365
and white in little
slips of paper in a bowl

00:01:05.365 --> 00:01:07.323
and they're just going
to pick one of them out.

00:01:07.323 --> 00:01:09.227
So all of these
are equally likely.

00:01:09.227 --> 00:01:10.810
So given this, that
they're just going

00:01:10.810 --> 00:01:12.226
to flip a coin to
pick the engine,

00:01:12.226 --> 00:01:16.100
and that all of these, the
colors all equally likely,

00:01:16.100 --> 00:01:17.970
I want to think
about the probability

00:01:17.970 --> 00:01:20.130
of getting a
six-cylinder white car.

00:01:25.299 --> 00:01:26.840
So I encourage you
to pause the video

00:01:26.840 --> 00:01:29.070
and think about it on your own.

00:01:29.070 --> 00:01:30.520
Well, one way to
think about this

00:01:30.520 --> 00:01:32.640
is, well, what are
all the equally likely

00:01:32.640 --> 00:01:34.300
possible outcomes?

00:01:34.300 --> 00:01:38.020
And then which of those
match six-cylinder white car?

00:01:38.020 --> 00:01:40.920
Well, first, we could think
about the engine decision.

00:01:40.920 --> 00:01:43.580
We're either going to get
a four-cylinder engine.

00:01:43.580 --> 00:01:45.370
So the first decision
is the engine.

00:01:45.370 --> 00:01:46.170
You could view it that way.

00:01:46.170 --> 00:01:48.253
You're either going to get
a four-cylinder engine,

00:01:48.253 --> 00:01:51.000
or you're going to get
a six-cylinder engine.

00:01:51.000 --> 00:01:53.350
Now, if you got a
four-cylinder engine,

00:01:53.350 --> 00:02:04.890
you're either going to get
red, blue, green, or white.

00:02:07.774 --> 00:02:09.440
And if you've got a
six-cylinder engine,

00:02:09.440 --> 00:02:13.220
once again, you're
either going to get red,

00:02:13.220 --> 00:02:15.480
blue-- I think you see
where this is going.

00:02:15.480 --> 00:02:17.080
That's not blue.

00:02:17.080 --> 00:02:26.835
Red, blue, green, or white.

00:02:29.800 --> 00:02:33.060
So how many possible
outcomes are there?

00:02:33.060 --> 00:02:34.505
Well, you could just count.

00:02:34.505 --> 00:02:35.880
You could kind of
say, the leaves

00:02:35.880 --> 00:02:39.710
of this tree diagram-- one, two,
three, four, five, six, seven,

00:02:39.710 --> 00:02:41.010
eight possible outcomes.

00:02:41.010 --> 00:02:42.240
And that makes sense.

00:02:42.240 --> 00:02:45.702
You have two possible engines
times four possible colors.

00:02:45.702 --> 00:02:47.160
You see that right
here-- one group

00:02:47.160 --> 00:02:49.330
of four, two groups of four.

00:02:49.330 --> 00:02:53.530
So this outcome right here
is a four-cylinder blue car.

00:02:53.530 --> 00:02:57.720
And this outcome over here
is a six-cylinder green car.

00:02:57.720 --> 00:02:59.950
So there's eight equally
possible outcomes.

00:02:59.950 --> 00:03:02.400
And which outcome matches
the one that we, I guess,

00:03:02.400 --> 00:03:04.427
are hoping for, the
white six-cylinder car?

00:03:04.427 --> 00:03:06.010
Well, that's this
one right over here.

00:03:06.010 --> 00:03:08.420
It's one of eight
equally likely events.

00:03:08.420 --> 00:03:10.427
So we have a 1/8 probability.

00:03:10.427 --> 00:03:12.010
This wasn't the only
way that we could

00:03:12.010 --> 00:03:13.596
have drawn the tree diagram.

00:03:13.596 --> 00:03:14.970
We could have
thought about color

00:03:14.970 --> 00:03:16.812
as the first row of this tree.

00:03:16.812 --> 00:03:18.520
So we could have said,
look, we're either

00:03:18.520 --> 00:03:20.290
going to get a-- let
me do it down here,

00:03:20.290 --> 00:03:22.330
so I have a little more
space-- we're either

00:03:22.330 --> 00:03:28.710
going to get a red, a
blue-- that's not blue.

00:03:28.710 --> 00:03:31.150
Changing colors
is the hard part.

00:03:31.150 --> 00:03:44.520
A blue, a green, or a white car.

00:03:44.520 --> 00:03:46.080
And then for each
of those colors,

00:03:46.080 --> 00:03:47.910
I'm either going to get a
four-cylinder or a six-cylinder

00:03:47.910 --> 00:03:48.650
engine.

00:03:48.650 --> 00:03:51.135
So it's either going
to be four or six.

00:04:00.800 --> 00:04:05.680
This would be another way
of drawing a tree diagram

00:04:05.680 --> 00:04:07.604
to represent all
of the outcomes.

00:04:07.604 --> 00:04:09.270
So what is this outcome
right over here?

00:04:09.270 --> 00:04:11.390
This is a six-cylinder red car.

00:04:11.390 --> 00:04:15.000
This is a four-cylinder
blue car right over here.

00:04:15.000 --> 00:04:16.529
Which is the one
that we care about?

00:04:16.529 --> 00:04:17.930
White six-cylinder car?

00:04:17.930 --> 00:04:19.490
That's this outcome
right over here.

00:04:19.490 --> 00:04:22.610
Once again, you see you have
eight equally likely outcomes.

00:04:22.610 --> 00:04:25.340
And that happens because you
have four possible colors.

00:04:25.340 --> 00:04:27.390
And for each of those
four possible colors,

00:04:27.390 --> 00:04:30.770
you have two different
engine types.


Construct a triangle with constraints

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.010
If someone walks up to you
on the street and says,

00:00:03.010 --> 00:00:05.260
all right, I have a
challenge for you.

00:00:05.260 --> 00:00:11.986
I want to construct a triangle
that has sides of length 2.

00:00:11.986 --> 00:00:15.920
So sides of length-- let me
write this a little bit neater.

00:00:15.920 --> 00:00:21.050
Sides of length 2, 2, and 5.

00:00:21.050 --> 00:00:23.190
Can you do this?

00:00:23.190 --> 00:00:25.050
Well, let's try to do it.

00:00:25.050 --> 00:00:29.080
And we'll start with the longest
side, the side of length 5.

00:00:29.080 --> 00:00:32.200
So the side of length 5.

00:00:32.200 --> 00:00:34.250
That's that side
right over there.

00:00:34.250 --> 00:00:37.070
And now, let's try to draw
the sides of length 2.

00:00:37.070 --> 00:00:38.570
Every side on a
triangle, obviously,

00:00:38.570 --> 00:00:39.861
connects with every other side.

00:00:39.861 --> 00:00:42.260
So that's one side of length 2.

00:00:42.260 --> 00:00:44.940
And then this is another
side of lengths 2.

00:00:44.940 --> 00:00:46.125
Another side of length 2.

00:00:46.125 --> 00:00:48.500
And you might say, fine, these
aren't touching right now,

00:00:48.500 --> 00:00:49.430
these two points.

00:00:49.430 --> 00:00:51.555
In order to make a triangle,
we have to touch them.

00:00:51.555 --> 00:00:53.280
So let me move them
closer to each other.

00:00:53.280 --> 00:00:55.320
But we have to remember, we
have to keep these side lengths

00:00:55.320 --> 00:00:56.110
the same.

00:00:56.110 --> 00:00:58.570
And we have to keep touching
the side of length 5

00:00:58.570 --> 00:00:59.720
at its endpoint.

00:00:59.720 --> 00:01:01.510
So we could try to move them in.

00:01:01.510 --> 00:01:04.769
We could try to move them in,
but what's going to happen?

00:01:04.769 --> 00:01:07.070
Well, you could rotate
them all the way down

00:01:07.070 --> 00:01:10.980
and they're still not going
to touch because 2 plus 2

00:01:10.980 --> 00:01:12.259
is still not equal to 5.

00:01:12.259 --> 00:01:13.800
They rotate all the
way down, they're

00:01:13.800 --> 00:01:15.640
still going to be 1 apart.

00:01:15.640 --> 00:01:18.610
So you cannot construct
this triangle.

00:01:18.610 --> 00:01:20.440
You cannot construct
this triangle.

00:01:20.440 --> 00:01:24.950
And I think you're noticing
a property of triangles.

00:01:24.950 --> 00:01:28.760
The longest side cannot
be longer than the sum

00:01:28.760 --> 00:01:30.590
of the other two sides.

00:01:30.590 --> 00:01:33.160
Here, the sum of the
other two sides is 4.

00:01:33.160 --> 00:01:34.280
2 plus 2 is 4.

00:01:34.280 --> 00:01:36.030
And the other side is longer.

00:01:36.030 --> 00:01:37.740
And even if the other
side was exactly

00:01:37.740 --> 00:01:39.450
equal to the sum of
the other two sides,

00:01:39.450 --> 00:01:41.440
you're going to have
a degenerate triangle.

00:01:41.440 --> 00:01:42.500
Let me draw that.

00:01:42.500 --> 00:01:47.140
So this would be side,
say, 2, 2, and 4.

00:01:47.140 --> 00:01:49.610
So let's draw the
side of length 4.

00:01:49.610 --> 00:01:52.130
Side of length 4.

00:01:52.130 --> 00:01:53.710
Side of length 4.

00:01:53.710 --> 00:01:57.420
Let me draw it a
little bit shorter.

00:01:57.420 --> 00:01:59.960
So that's your side of length 4.

00:01:59.960 --> 00:02:03.185
And then, in order to make the
two sides of length 2 touch,

00:02:03.185 --> 00:02:06.230
in order to make them touch,
you have to rotate them

00:02:06.230 --> 00:02:09.050
all the way inward You
have to rotate them

00:02:09.050 --> 00:02:12.650
all the way inward so that
both this angle and this angle

00:02:12.650 --> 00:02:15.010
essentially have to
become 0 degrees.

00:02:15.010 --> 00:02:17.260
And so your resulting triangle,
if you rotate this one

00:02:17.260 --> 00:02:18.880
all the way in and you
rotate this all the way in,

00:02:18.880 --> 00:02:20.590
the points will actually touch.

00:02:20.590 --> 00:02:22.840
But this triangle will
have no area anymore.

00:02:22.840 --> 00:02:26.190
This will become a
degenerate triangle.

00:02:26.190 --> 00:02:28.787
And it really looks more
like a line segment.

00:02:28.787 --> 00:02:29.870
So let me write that down.

00:02:29.870 --> 00:02:32.420
This is a degenerate.

00:02:32.420 --> 00:02:35.440
In order for you to draw
a non-degenerate triangle,

00:02:35.440 --> 00:02:37.470
the sum of the
other two sides have

00:02:37.470 --> 00:02:40.600
to be longer than
the longest side.

00:02:40.600 --> 00:02:44.160
So for example, you could
definitely draw a triangle

00:02:44.160 --> 00:02:47.790
with sides of
length 3, 3, and 5.

00:02:47.790 --> 00:02:52.439
So if that's the side of
length 5, and then this--

00:02:52.439 --> 00:02:53.980
if you were to rotate
all the way in,

00:02:53.980 --> 00:03:00.070
those two points would-- let me
draw this a little bit neater.

00:03:02.900 --> 00:03:04.794
So let's say that's
where they connect.

00:03:04.794 --> 00:03:06.210
And we know that
we could do that,

00:03:06.210 --> 00:03:08.030
because if you think about it,
if you were to keep rotating

00:03:08.030 --> 00:03:10.680
these, they're going to pass
each other at some point.

00:03:10.680 --> 00:03:12.439
They're going to
have to overlap.

00:03:12.439 --> 00:03:14.230
If you tried to make
a degenerate triangle,

00:03:14.230 --> 00:03:15.396
these points wouldn't touch.

00:03:15.396 --> 00:03:18.200
They'd actually overlap by
one unit right over here.

00:03:18.200 --> 00:03:20.690
So you could rotate them
out and actually form

00:03:20.690 --> 00:03:22.730
a non-degenerate triangle.

00:03:22.730 --> 00:03:24.360
So this one, you
absolutely could.

00:03:24.360 --> 00:03:26.276
And then there's another
interesting question,

00:03:26.276 --> 00:03:28.440
is this the only triangle
that you could construct

00:03:28.440 --> 00:03:31.560
that has sides of
length 3, 3, and 5?

00:03:31.560 --> 00:03:33.810
Well, you can't
change this length.

00:03:33.810 --> 00:03:36.210
So you can't change that
point and that point.

00:03:36.210 --> 00:03:38.266
And then, you can't
change these two lengths.

00:03:38.266 --> 00:03:39.640
So the only place
where they will

00:03:39.640 --> 00:03:42.580
be able to touch each other is
going to be right over there.

00:03:42.580 --> 00:03:45.110
So this right over here
is the only triangle

00:03:45.110 --> 00:03:46.517
that meets those constraints.

00:03:46.517 --> 00:03:48.100
You could rotate it
and whatever else.

00:03:48.100 --> 00:03:50.930
But if you rotate this, it's
still the same triangle.

00:03:50.930 --> 00:03:53.480
This is the only triangle
that has sides of length 3, 3,

00:03:53.480 --> 00:03:54.230
and 5.

00:03:54.230 --> 00:03:55.720
You can't change
any of the angles

00:03:55.720 --> 00:03:58.177
somehow to get a
different triangle.


Construct a right isosceles triangle

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.060
They're asking us to
draw a right triangle.

00:00:03.060 --> 00:00:05.160
So that means it has to
have a 90-degree angle.

00:00:05.160 --> 00:00:07.280
But it's also an
isosceles triangle,

00:00:07.280 --> 00:00:10.010
so that means it has to have
at least two sides equal

00:00:10.010 --> 00:00:11.655
and has two sides of length 3.

00:00:11.655 --> 00:00:13.530
So those two sides that
are going to be equal

00:00:13.530 --> 00:00:15.150
are going to be of
length 3, and it's

00:00:15.150 --> 00:00:16.540
got to be a right triangle.

00:00:16.540 --> 00:00:18.120
So let's see if we can do that.

00:00:18.120 --> 00:00:21.410
So let's try to make this right
over here the right angle.

00:00:21.410 --> 00:00:27.340
And let's make this side and
this side have length 3, so 3

00:00:27.340 --> 00:00:31.310
and then 3 right over there.

00:00:31.310 --> 00:00:33.690
Let me make sure I get
that right angle right.

00:00:33.690 --> 00:00:34.510
OK, there you go.

00:00:34.510 --> 00:00:35.920
So it's a right angle.

00:00:35.920 --> 00:00:37.050
It's isosceles.

00:00:37.050 --> 00:00:39.240
At least two sides are equal.

00:00:39.240 --> 00:00:41.230
And the two sides have length 3.

00:00:41.230 --> 00:00:43.670
So it seems like we've met
all of our constraints.

00:00:43.670 --> 00:00:45.710
Now they say, is there
a unique triangle

00:00:45.710 --> 00:00:47.400
that satisfies this condition?

00:00:47.400 --> 00:00:49.280
So another way of
rephrasing that, is this

00:00:49.280 --> 00:00:51.010
the only triangle
that I could have

00:00:51.010 --> 00:00:53.250
drawn that meets
these conditions?

00:00:53.250 --> 00:00:53.750
Well.

00:00:53.750 --> 00:00:56.291
I can't change this angle if I
want to meet these conditions.

00:00:56.291 --> 00:00:57.850
I can't change
these two lengths.

00:00:57.850 --> 00:00:59.610
And if you keep
this angle constant

00:00:59.610 --> 00:01:00.790
and you keep these
two lengths constant,

00:01:00.790 --> 00:01:02.690
then this point and this
point are going to be

00:01:02.690 --> 00:01:03.740
there are no matter what.

00:01:03.740 --> 00:01:07.500
So this is the only side that
can connect those two points.

00:01:07.500 --> 00:01:11.017
So this is the only triangle
that meets those conditions.

00:01:11.017 --> 00:01:12.600
You can't have
different side lengths,

00:01:12.600 --> 00:01:15.160
or you couldn't have different
angles right over here

00:01:15.160 --> 00:01:17.364
and also meet those conditions.

00:01:17.364 --> 00:01:18.780
So is there a
unique triangle that

00:01:18.780 --> 00:01:20.370
satisfies the given conditions?

00:01:20.370 --> 00:01:23.490
Yes, there's only
one unique triangle.


Find measure of angles in a word problem

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.020
There is a half
an apple pie left.

00:00:03.020 --> 00:00:06.240
You want to eat twice what
your little brother eats,

00:00:06.240 --> 00:00:09.290
but you also need to save
a slice for your mom.

00:00:09.290 --> 00:00:12.630
You can cut her a slice
that is 30 degrees.

00:00:12.630 --> 00:00:16.630
What is the measure of your
piece of the pie in degrees?

00:00:16.630 --> 00:00:19.030
So to tackle this, we just
have to remember a few things.

00:00:19.030 --> 00:00:21.580
We have to remember that
the degrees in a circle--

00:00:21.580 --> 00:00:24.000
so if we were to go all
the way around a circle

00:00:24.000 --> 00:00:28.070
that that would be 360 degrees.

00:00:28.070 --> 00:00:30.300
But we only have
half a pie here.

00:00:30.300 --> 00:00:32.870
And actually, let me draw
a little bit differently.

00:00:32.870 --> 00:00:36.050
So if we had a whole pie--
so if we started here,

00:00:36.050 --> 00:00:38.890
and we had a whole pie, we
went all the way around,

00:00:38.890 --> 00:00:41.000
that would be 360 degrees.

00:00:41.000 --> 00:00:42.780
But we only have a half pie.

00:00:42.780 --> 00:00:47.670
So we only have 180 degrees
and let me do that in a color

00:00:47.670 --> 00:00:50.020
you're more likely to see.

00:00:50.020 --> 00:00:51.290
That's hard to see as well.

00:00:51.290 --> 00:00:54.461
We only have 180
degrees of pie left,

00:00:54.461 --> 00:00:56.210
but let's just keep
that in mind and think

00:00:56.210 --> 00:00:59.210
about how we're going to
split it between ourselves,

00:00:59.210 --> 00:01:01.790
our brother, and our mom.

00:01:01.790 --> 00:01:04.129
So let's define some variables.

00:01:04.129 --> 00:01:09.990
Let's let x be equal to the
degree measure of my brother's

00:01:09.990 --> 00:01:12.280
pie or your little
brother's pie.

00:01:12.280 --> 00:01:19.270
So this is degree
measure of brother's pie.

00:01:22.700 --> 00:01:25.390
And then what would the
amount of pie you eat be?

00:01:25.390 --> 00:01:28.340
Well, it says you eat twice
what your little brother eats.

00:01:28.340 --> 00:01:34.850
So 2x would be equal what
you eat, and it's really

00:01:34.850 --> 00:01:38.130
the degree measure
of what you eat.

00:01:38.130 --> 00:01:39.760
And then how much
does your mom eat?

00:01:39.760 --> 00:01:41.895
Well, it says you cut her
slice that is 30 degrees.

00:01:41.895 --> 00:01:44.020
So your mom's going to get
a 30 degree slice, going

00:01:44.020 --> 00:01:46.050
to get a slice of
something like that.

00:01:46.050 --> 00:01:48.260
So the amount that
your brother eats,

00:01:48.260 --> 00:01:53.750
x, plus the amount that you eat,
plus the amount that your mom

00:01:53.750 --> 00:01:58.510
gets, plus 30 degrees, is going
to be equal to this half pie.

00:01:58.510 --> 00:02:00.840
And remember, all of
these are degree measures.

00:02:00.840 --> 00:02:04.600
So it's going to be
equal to 180 degrees.

00:02:04.600 --> 00:02:06.290
And just to visualize
it over here,

00:02:06.290 --> 00:02:09.150
let me draw it down here
where it's easier to see it.

00:02:09.150 --> 00:02:11.980
We have half a pie that
we're dealing with.

00:02:11.980 --> 00:02:16.930
We're going to save 30
degrees for our mom.

00:02:16.930 --> 00:02:19.877
So that's 30 degrees
right over there.

00:02:19.877 --> 00:02:21.585
Your brother is going
to eat some amount.

00:02:24.390 --> 00:02:26.152
So that is x.

00:02:26.152 --> 00:02:28.110
And then you're going to
eat twice that amount.

00:02:28.110 --> 00:02:31.520
So that is 2x. x is the
measure of this angle,

00:02:31.520 --> 00:02:33.900
and then 2x is a
measure of that angle.

00:02:33.900 --> 00:02:36.560
So you see that 30
degrees plus x plus 2x,

00:02:36.560 --> 00:02:39.190
or x plus 2x plus
30 degrees, is going

00:02:39.190 --> 00:02:41.970
to be equal to 180 degrees.

00:02:41.970 --> 00:02:44.080
Now, we can simplify this.

00:02:44.080 --> 00:02:46.890
If we have one of something and
then have another two of it,

00:02:46.890 --> 00:02:48.590
how much do we have now?

00:02:48.590 --> 00:02:50.310
Well, I now have three x's.

00:02:50.310 --> 00:02:53.150
So 1x plus 2x is going to be 3x.

00:02:53.150 --> 00:03:02.390
So I have 3x plus 30 degrees
is going to be equal to,

00:03:02.390 --> 00:03:03.730
of course, 180 degrees.

00:03:06.580 --> 00:03:08.990
Now, to solve for x, we
can subtract 30 degrees

00:03:08.990 --> 00:03:10.540
from both sides.

00:03:10.540 --> 00:03:14.530
So minus 30 degrees,
minus 30 degrees.

00:03:14.530 --> 00:03:16.980
And I could have just written--
I could have all assumed

00:03:16.980 --> 00:03:19.790
that I'm doing it in degrees and
then just done it at the end.

00:03:19.790 --> 00:03:22.290
I keep writing it so I'll
just keep going with that.

00:03:22.290 --> 00:03:29.330
And then we are left
with 3x equaling

00:03:29.330 --> 00:03:33.475
180 degrees minus 30 degrees
is equal to 150 degrees.

00:03:36.440 --> 00:03:41.270
And now we can just
divide both sides by 3,

00:03:41.270 --> 00:03:46.370
and we're left with x equaling
150 divided by 3 is 50 degrees.

00:03:46.370 --> 00:03:48.070
x is equal to 50 degrees.

00:03:48.070 --> 00:03:49.360
Now, we have to be careful.

00:03:49.360 --> 00:03:51.260
x is not what
they're asking for.

00:03:51.260 --> 00:03:53.980
They're asking for the
measure of your piece of pie

00:03:53.980 --> 00:03:55.060
in degrees.

00:03:55.060 --> 00:03:58.350
x is the degree measure of
your brother's piece of pie.

00:03:58.350 --> 00:04:00.770
What you eat is 2 times that.

00:04:00.770 --> 00:04:04.600
So if x is 50
degrees, 2 times that

00:04:04.600 --> 00:04:07.150
is going to be 100 degrees.

00:04:07.150 --> 00:04:10.580
So what is the measure of your
piece of the pie in degrees?

00:04:10.580 --> 00:04:14.310
It is going to be 100 degrees.

00:04:14.310 --> 00:04:18.769
So if we draw our pie again,
if we draw our half pie,

00:04:18.769 --> 00:04:20.570
you have 30 degrees
for your mom.

00:04:20.570 --> 00:04:22.770
So that's 30 degrees
for your mom.

00:04:22.770 --> 00:04:24.690
You have 50 degrees
for your brother.

00:04:28.110 --> 00:04:33.350
And then you have twice that
for yourself, 100 degrees.


Find measure of vertical angles

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.520
So we have two
intersecting lines here,

00:00:02.520 --> 00:00:04.637
and then we have this
other purple-looking line.

00:00:04.637 --> 00:00:06.220
And then they've
given us some angles.

00:00:06.220 --> 00:00:08.920
They tell us that the measure
of this angle right over here

00:00:08.920 --> 00:00:10.010
is 7x.

00:00:10.010 --> 00:00:13.020
The measure of this angle
right over here is 60 degrees.

00:00:13.020 --> 00:00:17.240
And the measure of this
angle right over here is x.

00:00:17.240 --> 00:00:20.300
So let's try to figure out
what all of these angles are.

00:00:20.300 --> 00:00:23.410
So to do that, we have
to figure out what x is.

00:00:23.410 --> 00:00:27.570
And there's a big clue here,
because the 60-degree angle

00:00:27.570 --> 00:00:30.440
plus the x angle,
they're adjacent.

00:00:30.440 --> 00:00:33.610
And if you add these two angles
together, their outer rays,

00:00:33.610 --> 00:00:36.299
our vertical angle
with this 7x angle.

00:00:36.299 --> 00:00:38.840
So we could say-- and just to
visualize that a little better,

00:00:38.840 --> 00:00:41.060
let me color it in.

00:00:41.060 --> 00:00:43.540
Actually, let me do it this way.

00:00:43.540 --> 00:00:48.080
You see that this
angle out here--

00:00:48.080 --> 00:00:49.970
let me do it in a color
I haven't used yet.

00:00:49.970 --> 00:00:52.530
This entire angle
over here, which

00:00:52.530 --> 00:00:54.960
is going to be 60
degrees plus x, that's

00:00:54.960 --> 00:00:57.210
a vertical angle with
this angle, the one that

00:00:57.210 --> 00:00:58.850
has measure 7x.

00:00:58.850 --> 00:01:02.980
So we could say that 60
degrees plus x is equal to 7x

00:01:02.980 --> 00:01:05.340
because vertical
angles are equal.

00:01:05.340 --> 00:01:06.890
So let's write that down.

00:01:06.890 --> 00:01:08.310
We get 60.

00:01:08.310 --> 00:01:11.030
And we'll assume that
everything is in degrees.

00:01:11.030 --> 00:01:14.770
60 plus-- let me do that
in this other color.

00:01:14.770 --> 00:01:20.180
60 plus x is going
to be equal to 7x.

00:01:20.180 --> 00:01:23.195
And now we just
have to solve for x.

00:01:23.195 --> 00:01:24.570
So the simplest
thing to do would

00:01:24.570 --> 00:01:27.320
be to get all over x's on
one side of the equation.

00:01:27.320 --> 00:01:29.820
I've already gotten seven
x's on this right-hand side,

00:01:29.820 --> 00:01:33.594
so let's get rid of all of
the x's on the left-hand side.

00:01:33.594 --> 00:01:35.260
And the easiest way
to get rid of this x

00:01:35.260 --> 00:01:37.740
is to subtract x from
the left-hand side.

00:01:37.740 --> 00:01:40.100
But of course, in order
to keep it an equation,

00:01:40.100 --> 00:01:41.750
we can't just do
something to one side.

00:01:41.750 --> 00:01:43.291
Otherwise, it won't
be equal anymore.

00:01:43.291 --> 00:01:45.020
We have to do it to both sides.

00:01:45.020 --> 00:01:47.280
So let's subtract
x from both sides.

00:01:47.280 --> 00:01:52.190
And on the left-hand side,
we are left with just the 60.

00:01:52.190 --> 00:01:54.020
So we're left with just a 60.

00:01:54.020 --> 00:01:57.360
And then that is going to
be equal to 7x minus x.

00:01:57.360 --> 00:02:00.340
If I have 7 of something
and I get rid of 1 of them,

00:02:00.340 --> 00:02:02.650
I'm going to have 6 of
that something left.

00:02:02.650 --> 00:02:05.140
So that's going
to be equal to 6x.

00:02:05.140 --> 00:02:07.610
So we have 6 times
something is equal to 60.

00:02:07.610 --> 00:02:09.979
You could probably figure
that out in your head.

00:02:09.979 --> 00:02:11.850
But I will do it a
little systematically.

00:02:11.850 --> 00:02:15.190
We can divide both sides
by 6 to solve for x.

00:02:15.190 --> 00:02:16.600
So let's do that.

00:02:16.600 --> 00:02:19.660
And we would be left
with x is equal to.

00:02:19.660 --> 00:02:22.780
60 divided by 6 is 10.

00:02:22.780 --> 00:02:25.419
And we reminded ourselves that
everything was in degrees.

00:02:25.419 --> 00:02:26.710
And we could even do that here.

00:02:26.710 --> 00:02:27.680
This was in degrees.

00:02:27.680 --> 00:02:28.940
This is in degrees.

00:02:28.940 --> 00:02:31.270
And so this is in
degrees right over here.

00:02:31.270 --> 00:02:34.830
So the measure of this angle
right over here is 10 degrees.

00:02:34.830 --> 00:02:38.010
So this one right over
there is 10 degrees.

00:02:38.010 --> 00:02:40.100
This is, of course, 60 degrees.

00:02:40.100 --> 00:02:42.260
You add them together, 60
degrees plus 10 degrees

00:02:42.260 --> 00:02:43.780
is 70 degrees.

00:02:43.780 --> 00:02:47.081
So this bigger angle right
over here is 70 degrees.

00:02:47.081 --> 00:02:49.455
And of course, this one over
here, it's a vertical angle.

00:02:49.455 --> 00:02:50.871
It's going to have
to be the same.

00:02:50.871 --> 00:02:52.520
And we see that, 7 times x.

00:02:52.520 --> 00:02:59.244
7 times 10 degrees is
70 degrees as well.


Find the volume of rectangles inside rectangles

WEBVTT
Kind: captions
Language: en

00:00:00.980 --> 00:00:04.010
The Ubas are moving
from Houston to Egypt.

00:00:04.010 --> 00:00:06.260
They pack their belongings
in rectangular crates

00:00:06.260 --> 00:00:11.270
and hire a boxcar to ship the
crates across land and sea.

00:00:11.270 --> 00:00:14.740
The crates are made specifically
to fit inside the boxcar

00:00:14.740 --> 00:00:17.050
with their bases facing down.

00:00:17.050 --> 00:00:21.850
Each crate has a base 5 meters
long by 1.5 meters wide.

00:00:21.850 --> 00:00:22.760
So let me draw that.

00:00:22.760 --> 00:00:30.120
So crate is 5 meters
long, and 1.5 meters wide,

00:00:30.120 --> 00:00:32.060
and has a height of 2 meters.

00:00:32.060 --> 00:00:34.830
So its height might look
something like this.

00:00:34.830 --> 00:00:36.200
So it has a height of 2 meters.

00:00:36.200 --> 00:00:38.120
So that's each of the crates.

00:00:38.120 --> 00:00:42.120
And they're designed
to fit inside a boxcar.

00:00:42.120 --> 00:00:43.620
So this is a crate
right over here.

00:00:43.620 --> 00:00:45.554
I'll do my best to draw a crate.

00:00:45.554 --> 00:00:47.470
And they give us the
dimensions of the boxcar.

00:00:47.470 --> 00:00:50.610
A boxcar is 15 meters long.

00:00:50.610 --> 00:00:53.110
So let me draw a boxcar here.

00:00:53.110 --> 00:00:55.640
So it's 15 meters long.

00:00:55.640 --> 00:00:58.560
Maybe I'll try to make sure
I can fit it on the page.

00:00:58.560 --> 00:01:01.010
So that this whole
distance would be 5 meters,

00:01:01.010 --> 00:01:05.480
and then another 5 meters here,
and then another 5 meters here.

00:01:05.480 --> 00:01:08.040
So that would be 15 meters long.

00:01:08.040 --> 00:01:13.420
So you could fit three of the
crates along an edge like that.

00:01:16.680 --> 00:01:20.720
And then, they tell us
that it is 3 meters wide.

00:01:20.720 --> 00:01:22.310
So this is 1.5 meters wide.

00:01:22.310 --> 00:01:27.450
So you could put two of
these to get you to 3 meters.

00:01:27.450 --> 00:01:30.130
Let me draw this so
we can see what's

00:01:30.130 --> 00:01:32.570
going on behind the scenes.

00:01:32.570 --> 00:01:35.960
So you could go 3 meters
wide for a boxcar,

00:01:35.960 --> 00:01:37.550
and then it is 4 meters high.

00:01:37.550 --> 00:01:40.830
So each of these
are 2 meters high.

00:01:40.830 --> 00:01:44.690
So you could stack one more.

00:01:44.690 --> 00:01:47.190
And so you have 2
meters plus 2 meters.

00:01:47.190 --> 00:01:53.350
This entire distance right over
here is going to be 4 meters.

00:01:53.350 --> 00:01:58.687
And I could draw the rest
of the boxcar like this.

00:01:58.687 --> 00:02:01.270
So there's a couple of ways to
think about how many crates you

00:02:01.270 --> 00:02:02.570
could fit in a boxcar.

00:02:02.570 --> 00:02:05.070
One way would be just the way
that we're doing it right now.

00:02:05.070 --> 00:02:06.310
We could visualize.

00:02:06.310 --> 00:02:09.009
How many can you fit in this
direction along the length?

00:02:09.009 --> 00:02:11.039
How many can you
fit along the width?

00:02:11.039 --> 00:02:13.250
And how many can you
fit along the height?

00:02:13.250 --> 00:02:15.571
And essentially, if we
multiply those three numbers,

00:02:15.571 --> 00:02:17.320
we would have counted
the number of crates

00:02:17.320 --> 00:02:18.640
that could fit inside.

00:02:18.640 --> 00:02:23.150
So you could fit 1,
2, 3 along the length.

00:02:23.150 --> 00:02:24.450
So that'd be 3.

00:02:24.450 --> 00:02:26.040
You could fit 2 along the width.

00:02:26.040 --> 00:02:30.210
1.5 and 1.5 gets you to
3 meters, so times 2.

00:02:30.210 --> 00:02:34.160
And then you could fit 2
along the height, so times 2,

00:02:34.160 --> 00:02:38.060
gets us to 3 times 2
is 6 times 2 is 12.

00:02:38.060 --> 00:02:41.190
You can fit 12
crates in the boxcar.

00:02:41.190 --> 00:02:44.070
Now, another way you could have
done it is you could say, OK,

00:02:44.070 --> 00:02:46.664
they're telling us that
these are designed to fit.

00:02:46.664 --> 00:02:48.580
So we really just have
to compare the volumes.

00:02:48.580 --> 00:02:50.620
How many times
more is the volume

00:02:50.620 --> 00:02:52.410
of the box car than the crate?

00:02:52.410 --> 00:02:54.160
I like doing it
this way more, just

00:02:54.160 --> 00:02:56.580
to make sure that the
dimensions actually work out,

00:02:56.580 --> 00:02:58.510
so that you could
actually squeeze these in.

00:02:58.510 --> 00:03:00.180
Because if the
dimensions aren't right,

00:03:00.180 --> 00:03:03.994
even if the boxcar is 12
times the volume of one

00:03:03.994 --> 00:03:05.410
of the crates, if
the crates don't

00:03:05.410 --> 00:03:07.076
have the right
dimensions, you might not

00:03:07.076 --> 00:03:09.989
be able to squeeze exactly
12 crates in there.

00:03:09.989 --> 00:03:12.280
But they're telling us that
it is the exact dimensions.

00:03:12.280 --> 00:03:14.830
So we could figure out the
dimensions of the boxcar, then

00:03:14.830 --> 00:03:16.090
the dimensions of the crate.

00:03:16.090 --> 00:03:18.580
And then we could figure out
how much larger the boxcar is,

00:03:18.580 --> 00:03:21.510
how many times larger.

00:03:21.510 --> 00:03:23.270
Let's do the boxcar
in this blue color.

00:03:23.270 --> 00:03:27.950
The boxcar is 15 meters long, 3
meters wide, and 4 meters high.

00:03:27.950 --> 00:03:36.430
So boxcar volume is equal
to 15 in cubic meters.

00:03:36.430 --> 00:03:41.330
So there's 15 meters times
3 meters times 4 meters.

00:03:41.330 --> 00:03:43.350
So this is going to
be in cubic meters.

00:03:43.350 --> 00:03:47.170
So this is going to be--
let's see, 15 times 3 is 45.

00:03:47.170 --> 00:03:52.940
45 times 4 is 180 cubic meters.

00:03:52.940 --> 00:03:54.400
That's the boxcar volume.

00:03:54.400 --> 00:03:56.600
And then what's the
volume of the crate?

00:03:56.600 --> 00:04:02.140
Well, the crate volume--
if we do our math right,

00:04:02.140 --> 00:04:03.640
it should come out
to 1/12 of this,

00:04:03.640 --> 00:04:05.306
because that's what
we just figured out,

00:04:05.306 --> 00:04:08.530
is 5 times 1.5 times 2.

00:04:08.530 --> 00:04:13.800
So 5 times 1.5 times 2.

00:04:13.800 --> 00:04:20.610
Well, 1.5 times 2 is 3, times
5 is 15, so 15 cubic meters.

00:04:20.610 --> 00:04:23.460
So how many times larger is
the boxcar than the crate?

00:04:23.460 --> 00:04:26.340
Well, what's 180 divided by 15?

00:04:26.340 --> 00:04:28.770
Well, it's exactly 12.

00:04:28.770 --> 00:04:30.790
10 times 15 is 150.

00:04:30.790 --> 00:04:32.470
And then 2 times 15 is 30.

00:04:32.470 --> 00:04:34.800
150 plus 30 is 180.

00:04:34.800 --> 00:04:36.960
So notice, 180
divided by 15 is 12.

00:04:36.960 --> 00:04:38.760
So either way, however
you think about it--

00:04:38.760 --> 00:04:40.850
I find this one to be a
little bit easier to kind

00:04:40.850 --> 00:04:42.580
of just visualize
the boxes-- you

00:04:42.580 --> 00:04:45.960
can fit 12 crates in the boxcar.


Find the volume of a ring

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.200
Jamie wants to know the
volume of his gold ring

00:00:03.200 --> 00:00:04.950
in cubic inches.

00:00:04.950 --> 00:00:08.850
He gets a rectangular glass
with base 3 inches by 2 inches.

00:00:08.850 --> 00:00:11.820
So you see that here, the
base is 3 inches by 2 inches.

00:00:11.820 --> 00:00:14.150
And he fills the glass 4
inches high with water.

00:00:14.150 --> 00:00:17.010
So you see that over here,
4 inches high with water.

00:00:17.010 --> 00:00:19.060
Jamie drops his gold
ring in the glass

00:00:19.060 --> 00:00:22.970
and measures the new height of
the water to be 4.25 inches.

00:00:22.970 --> 00:00:25.170
So this is after the
gold ring is dropped.

00:00:25.170 --> 00:00:29.030
What is the volume of
Jamie's ring in cubic inches?

00:00:29.030 --> 00:00:31.750
Well when you start with
this water right over here

00:00:31.750 --> 00:00:35.980
and you add his ring, whatever
that volume is of his ring

00:00:35.980 --> 00:00:39.510
is going to displace an equal
volume of water and push it up.

00:00:39.510 --> 00:00:42.070
And so the incremental
volume that you now have

00:00:42.070 --> 00:00:44.960
is essentially going to
be the volume of his ring.

00:00:44.960 --> 00:00:47.965
Well what is the
incremental volume here?

00:00:47.965 --> 00:00:49.340
Well it's going
to be the volume.

00:00:49.340 --> 00:00:51.782
If you think about going
from this before volume

00:00:51.782 --> 00:00:53.240
to the after volume,
the difference

00:00:53.240 --> 00:00:56.760
is the base stays the same.

00:00:56.760 --> 00:00:59.560
It's 3 inches by 2
inches, the difference

00:00:59.560 --> 00:01:04.920
is-- to make it a little bit
neater-- the base is the same.

00:01:04.920 --> 00:01:07.040
The difference is the height.

00:01:07.040 --> 00:01:11.510
The height now is 4.25 inches
after dropping in the ring

00:01:11.510 --> 00:01:16.740
So the water went
up by 0.25 inches.

00:01:16.740 --> 00:01:21.444
Let me write that, 0.25 inches
is what the water went up by.

00:01:21.444 --> 00:01:22.860
So we could just
think about, what

00:01:22.860 --> 00:01:24.860
is this incremental
volume going to be?

00:01:28.820 --> 00:01:30.690
So this incremental
volume right over here,

00:01:30.690 --> 00:01:34.090
that I'm shading in with purple.

00:01:34.090 --> 00:01:36.210
Well to figure that out
we just have to measure.

00:01:36.210 --> 00:01:38.800
We just have to
multiply the length

00:01:38.800 --> 00:01:43.420
times the width times
the height times 0.25.

00:01:43.420 --> 00:01:51.360
So it's just going to
be 3 times 2 times 0.25.

00:01:51.360 --> 00:01:57.497
3 times 2 is 6,
times 0.25, and you

00:01:57.497 --> 00:01:59.580
could do that either on
paper or you might be able

00:01:59.580 --> 00:02:00.455
do that in your head.

00:02:00.455 --> 00:02:04.730
4 times 0.25 is going to
be 1, and you have 2 more

00:02:04.730 --> 00:02:07.430
times 0.25, that's
going to be 0.5.

00:02:07.430 --> 00:02:10.850
So this is going to be 1.50.

00:02:10.850 --> 00:02:13.530
And we multiply it inches
times inches times inches.

00:02:13.530 --> 00:02:17.130
So this is going to be
in terms of cubic inches.

00:02:17.130 --> 00:02:24.980
1.5 cubic inches is the
volume of Jamie's ring, which

00:02:24.980 --> 00:02:28.410
is actually a pretty sizable
volume for a gold ring.

00:02:28.410 --> 00:02:32.470
Maybe he has a very big finger
or he just likes to spend,

00:02:32.470 --> 00:02:34.880
or I guess is his,
whoever bought him

00:02:34.880 --> 00:02:38.950
the ring likes to
spend a lot on gold.


Impact of a radius change on the area of a circle

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.530
What I want to do
in this video is

00:00:01.530 --> 00:00:04.030
think about how does
the circumference

00:00:04.030 --> 00:00:06.180
and how does the area
of a circle change

00:00:06.180 --> 00:00:07.317
as we change its radius.

00:00:07.317 --> 00:00:08.900
And in particular,
we'll focus on what

00:00:08.900 --> 00:00:11.080
happens when we
double its radius.

00:00:11.080 --> 00:00:14.020
So let's think about a
circle right over here.

00:00:14.020 --> 00:00:15.860
So this is a circle.

00:00:15.860 --> 00:00:20.500
And let's say its radius is x
units, whatever our units is.

00:00:20.500 --> 00:00:23.350
So this distance
right over here is x.

00:00:23.350 --> 00:00:25.310
And then let's think
about another circle that

00:00:25.310 --> 00:00:26.490
has twice the radius.

00:00:26.490 --> 00:00:28.585
So its radius will be 2x.

00:00:28.585 --> 00:00:31.270
And so let me draw
its radius first,

00:00:31.270 --> 00:00:33.560
so it looks roughly accurate.

00:00:33.560 --> 00:00:37.040
So this is 2x is
this circle's radius.

00:00:37.040 --> 00:00:40.700
And so this circle might
look something like that.

00:00:40.700 --> 00:00:43.960
That's my best attempt at
freehand drawing a circle.

00:00:43.960 --> 00:00:46.750
So let's just think about
what the circumference of both

00:00:46.750 --> 00:00:50.070
of these are and what the
areas of both of these are.

00:00:50.070 --> 00:00:52.250
So the circumference
of any circle

00:00:52.250 --> 00:00:54.220
is 2 pi times the radius.

00:00:54.220 --> 00:00:56.560
So in this case, the
circumference-- and I'll

00:00:56.560 --> 00:01:01.430
use C for circumference-- is
equal to 2 pi times the radius,

00:01:01.430 --> 00:01:03.970
which in this case is x.

00:01:03.970 --> 00:01:05.519
Well, what's the
circumference here?

00:01:05.519 --> 00:01:07.260
Well, once again,
the circumference

00:01:07.260 --> 00:01:09.930
is equal to 2 pi
times the radius.

00:01:09.930 --> 00:01:13.050
But this time, the radius is 2x.

00:01:13.050 --> 00:01:16.480
So the circumference is equal
to 2 times pi times 2 times

00:01:16.480 --> 00:01:19.970
x, which is the same thing as
2 times 2 times pi times x.

00:01:19.970 --> 00:01:24.380
Or we could write it as 4 pi x.

00:01:24.380 --> 00:01:27.035
So we see here that this
circumference is twice

00:01:27.035 --> 00:01:27.910
as large as this one.

00:01:27.910 --> 00:01:34.380
To go from 2 pi x to 4 pi x,
you have to multiply by 2.

00:01:34.380 --> 00:01:35.480
So you doubled the radius.

00:01:35.480 --> 00:01:37.580
It doubled the circumference.

00:01:37.580 --> 00:01:39.170
Now, what about the area?

00:01:39.170 --> 00:01:41.400
And I'll do area in a new color.

00:01:41.400 --> 00:01:45.380
So we already know that area
is equal to pi r squared.

00:01:45.380 --> 00:01:48.270
In this circle, the
radius is of length x.

00:01:48.270 --> 00:01:51.070
So it's pi times x squared.

00:01:51.070 --> 00:01:53.800
In this circle right
over here, the area

00:01:53.800 --> 00:01:57.250
is going to be equal to pi
times the radius squared.

00:01:57.250 --> 00:02:01.026
But now the radius
is 2x-- 2x squared.

00:02:01.026 --> 00:02:02.650
Now, what is this
going to be equal to?

00:02:02.650 --> 00:02:07.350
Well, our area is equal to
pi 2x squared is 2x times 2x,

00:02:07.350 --> 00:02:11.670
which is the same
thing as 4x squared.

00:02:11.670 --> 00:02:19.830
Or we could rewrite this as
area is equal to 4 pi x squared.

00:02:19.830 --> 00:02:23.840
Notice, now the area has
increased not by a factor of 2.

00:02:23.840 --> 00:02:28.380
The area has increased
by a factor of 4

00:02:28.380 --> 00:02:30.039
when we doubled the radius.

00:02:30.039 --> 00:02:31.080
Now, why did this happen?

00:02:31.080 --> 00:02:34.100
And I encourage you to pause
the video and think about it.

00:02:34.100 --> 00:02:36.240
Well, it comes straight
out of the formulas

00:02:36.240 --> 00:02:38.070
for circumference and area.

00:02:38.070 --> 00:02:41.910
Remember, circumference
is equal to 2 pi

00:02:41.910 --> 00:02:49.020
r-- let me do this in
a different color--

00:02:49.020 --> 00:02:55.870
while area is equal
to pi r squared.

00:02:55.870 --> 00:02:58.670
So you see here, area is
proportional to the square

00:02:58.670 --> 00:03:00.252
of the radius.

00:03:00.252 --> 00:03:01.710
So if you double
this, you're going

00:03:01.710 --> 00:03:03.620
to increase your area
by a factor of 4.

00:03:03.620 --> 00:03:05.560
If you triple it, if
you triple your radius,

00:03:05.560 --> 00:03:07.770
you're going to increase
your area by a factor of 9.

00:03:07.770 --> 00:03:09.974
If you increase your
radius by a factor of 4,

00:03:09.974 --> 00:03:11.390
you're going to
increase your area

00:03:11.390 --> 00:03:14.900
by a factor of 4 squared or 16.

00:03:14.900 --> 00:03:16.870
While circumference--
whatever factor

00:03:16.870 --> 00:03:19.090
you increase your
radius, the circumference

00:03:19.090 --> 00:03:21.847
is going to increase
by that same factor.

00:03:21.847 --> 00:03:23.680
And if you don't believe
me-- we essentially

00:03:23.680 --> 00:03:25.930
showed it right here through
a little bit of algebra--

00:03:25.930 --> 00:03:30.100
but you could try it out with
as many numbers as you see fit.


Area of a shaded region

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.480
We're asked to find the area
of the shaded region, so

00:00:03.480 --> 00:00:06.380
the area of this
red-shaded region.

00:00:06.380 --> 00:00:07.920
So this is interesting.

00:00:07.920 --> 00:00:10.680
This is almost a
10 by 10 square,

00:00:10.680 --> 00:00:14.800
except we have these quarter
circles that are cut out.

00:00:14.800 --> 00:00:18.780
So the area of this would be the
area of what a 10 by 10 square

00:00:18.780 --> 00:00:22.500
would be minus the area
of these quarter circles.

00:00:22.500 --> 00:00:23.910
And each of these
quarter circles

00:00:23.910 --> 00:00:26.330
is a quarter of a
circle with a radius 3.

00:00:26.330 --> 00:00:28.110
I think we can assume
that all of these,

00:00:28.110 --> 00:00:29.550
if you took the
distance from here

00:00:29.550 --> 00:00:32.520
to the outside of this
quarter circle, have radius 3.

00:00:32.520 --> 00:00:34.980
So if you put four
quarter circles together,

00:00:34.980 --> 00:00:37.590
you're going to have a
complete white circle.

00:00:37.590 --> 00:00:39.060
So one way to
think about this is

00:00:39.060 --> 00:00:41.730
that the area of
this whole red region

00:00:41.730 --> 00:00:44.320
is going to be the area of
the entire square, which

00:00:44.320 --> 00:00:46.620
is 10 by 10.

00:00:46.620 --> 00:00:48.470
So it's going to be
10 times 10, which

00:00:48.470 --> 00:00:51.900
is 100 whatever
square units we have.

00:00:51.900 --> 00:00:53.660
And then we're going
to subtract out

00:00:53.660 --> 00:00:58.630
the area of the four
quarter circles.

00:00:58.630 --> 00:01:02.770
And that area is going to be
equivalent to the area of one

00:01:02.770 --> 00:01:04.660
circle with a radius of 3.

00:01:04.660 --> 00:01:08.110
So what's the area of
a circle with radius 3?

00:01:08.110 --> 00:01:10.840
Well, the formula
for area of a circle

00:01:10.840 --> 00:01:14.220
is pi r squared,
or r squared pi.

00:01:14.220 --> 00:01:15.400
So the radius is 3.

00:01:15.400 --> 00:01:22.380
So it's going to be 3 times 3,
which is 9, times pi-- 9 pi.

00:01:22.380 --> 00:01:28.660
So we have 100 minus 9 pi is
the area of the shaded region.

00:01:28.660 --> 00:01:30.740
And we got it right.


Circumference of a circle

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.550
If we know some circle
has an area of 36pi--

00:00:05.550 --> 00:00:09.700
so it has an area
of 36pi-- can we

00:00:09.700 --> 00:00:13.060
figure out what the
circumference of this circle

00:00:13.060 --> 00:00:13.580
is?

00:00:13.580 --> 00:00:15.340
And I encourage you
to pause this video,

00:00:15.340 --> 00:00:18.420
and try to think
about that question.

00:00:18.420 --> 00:00:21.850
Well, from the area, we could
figure out what the radius is,

00:00:21.850 --> 00:00:23.830
and then from that
radius, we can figure out

00:00:23.830 --> 00:00:25.410
what its circumference is.

00:00:25.410 --> 00:00:27.990
So we know that the
area, which is 36pi,

00:00:27.990 --> 00:00:32.530
is equal to pi r squared.

00:00:32.530 --> 00:00:35.110
And so if you look at it on
both sides of this equation,

00:00:35.110 --> 00:00:37.800
if we divide-- let me rewrite
it so it's a little bit clearer

00:00:37.800 --> 00:00:38.920
in a different color.

00:00:38.920 --> 00:00:45.280
So we could set up an equation
pi r squared is equal to 36pi.

00:00:45.280 --> 00:00:46.960
Now, if we want to
solve for the radius

00:00:46.960 --> 00:00:48.626
the first thing that
we might want to do

00:00:48.626 --> 00:00:51.590
is divide both sides by pi.

00:00:51.590 --> 00:00:57.430
Then, we're left with r
squared is equal to 36.

00:00:57.430 --> 00:01:00.024
Now, if we just solve this
as a pure math equation,

00:01:00.024 --> 00:01:02.440
you might say, OK, we could
take the positive and negative

00:01:02.440 --> 00:01:03.460
square root of 36.

00:01:03.460 --> 00:01:06.030
r could be plus
or minus 6, but we

00:01:06.030 --> 00:01:07.820
need to remember
that r is a distance,

00:01:07.820 --> 00:01:09.700
so we only care
about the positive.

00:01:09.700 --> 00:01:15.070
So if we take the principal root
of 36, we get r is equal to 6.

00:01:15.070 --> 00:01:18.740
From there, we can use this to
figure out the circumference.

00:01:18.740 --> 00:01:21.260
So the circumference
is equal to 2 pi r.

00:01:24.070 --> 00:01:28.460
Circumference is
equal to 2 pi r.

00:01:28.460 --> 00:01:30.250
And in this case,
r is equal to 6.

00:01:30.250 --> 00:01:36.500
So it's equal to 2
pi times 6, which

00:01:36.500 --> 00:01:40.970
is going to be equal to 12pi.

00:01:40.970 --> 00:01:44.290
So that's straightforward,
area 36pi, we leverage pi r

00:01:44.290 --> 00:01:46.420
squared to figure out
that the radius was 6,

00:01:46.420 --> 00:01:48.530
and then from that we
were able to figure out

00:01:48.530 --> 00:01:51.628
that the circumference was 12pi.


How to make a scale drawing

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.650
Cole is an urban planner.

00:00:02.650 --> 00:00:06.590
He wants to create a small
scale drawing of a city block.

00:00:06.590 --> 00:00:13.510
The block is a square with an
area of 8,100 square meters.

00:00:13.510 --> 00:00:15.860
Create a scale
drawing of the block

00:00:15.860 --> 00:00:20.034
using a scale factor of 0.1.

00:00:20.034 --> 00:00:21.700
So the first thing
we could think about,

00:00:21.700 --> 00:00:24.340
they give us the
area of the block.

00:00:24.340 --> 00:00:25.800
And it's a square block.

00:00:25.800 --> 00:00:27.940
So it has the same
length and width.

00:00:27.940 --> 00:00:30.400
So we could use that
information to figure out

00:00:30.400 --> 00:00:33.650
the length and
width of that block.

00:00:33.650 --> 00:00:36.770
So if it's a square, so let's
imagine a square block right

00:00:36.770 --> 00:00:38.090
over here.

00:00:38.090 --> 00:00:40.605
And that this is, I guess we
could call that the length.

00:00:40.605 --> 00:00:42.480
And then this is also
going to be the length.

00:00:42.480 --> 00:00:44.080
It's going to be
the same dimensions.

00:00:44.080 --> 00:00:45.538
We know that the
area is just going

00:00:45.538 --> 00:00:49.200
to be the length
times the length is

00:00:49.200 --> 00:00:53.450
going to be equal to 8,100.

00:00:53.450 --> 00:00:55.920
Or we could say that
our length squared

00:00:55.920 --> 00:00:59.000
is going to be equal to 8,100.

00:00:59.000 --> 00:01:03.590
So what times itself
is equal to 8,100?

00:01:03.590 --> 00:01:08.400
Well, the 81 might
jump out at you.

00:01:08.400 --> 00:01:14.150
We know that 9 times
9 is equal to 81.

00:01:14.150 --> 00:01:17.280
But then we have these two
zeroes right over here.

00:01:17.280 --> 00:01:20.010
But if we give each
of these nines a zero,

00:01:20.010 --> 00:01:22.410
then we'll end up with
two zeros in the product.

00:01:22.410 --> 00:01:28.520
So 90 times 90 is
equal to 8,100.

00:01:28.520 --> 00:01:32.610
So now we know the dimensions
of this square block.

00:01:32.610 --> 00:01:36.445
It's a 90 meter by 90
meter square block.

00:01:36.445 --> 00:01:41.440
It's my best attempt
to draw a square block.

00:01:41.440 --> 00:01:43.200
Now that's the
actual square block.

00:01:43.200 --> 00:01:45.670
Here let me draw it a little
bit more like a square.

00:01:45.670 --> 00:01:48.620
My first drawing looked a little
bit too much like a rhombus.

00:01:48.620 --> 00:01:49.490
So here you go.

00:01:49.490 --> 00:01:52.010
A little bit better
attempt at a square.

00:01:52.010 --> 00:01:54.940
Now we want to create at a
scale drawing of the block using

00:01:54.940 --> 00:01:57.540
a scale factor of 0.1.

00:01:57.540 --> 00:02:01.350
So the actual block, once again,
is 90 meters by 90 meters.

00:02:01.350 --> 00:02:03.130
But in our scale
drawing, and I'll

00:02:03.130 --> 00:02:07.150
do this in this purple
color, we essentially

00:02:07.150 --> 00:02:09.919
want each of the
dimensions to be

00:02:09.919 --> 00:02:15.020
1/10 of their
original dimensions.

00:02:15.020 --> 00:02:16.560
So we could take
the scale factor

00:02:16.560 --> 00:02:20.690
and multiply it by each of these
dimensions right over here.

00:02:20.690 --> 00:02:25.599
So 90 times 0.1, well
that's just going to be 9.

00:02:25.599 --> 00:02:26.890
This right over here is a 1/10.

00:02:26.890 --> 00:02:29.750
A 1/10 of 90 meters is
going to be 9 meters.

00:02:29.750 --> 00:02:33.130
And so this, when we draw the
scale drawing at the scale

00:02:33.130 --> 00:02:35.740
that Cole intends
to draw it at, we

00:02:35.740 --> 00:02:38.460
would draw 9 meters by 9 meters.

00:02:38.460 --> 00:02:40.060
And so let's go
to the actual tool

00:02:40.060 --> 00:02:44.680
and draw a 9 meter by 9 meter
square, or a 9 by 9 square.

00:02:44.680 --> 00:02:48.380
We can assume that the units
here are in terms of meters.

00:02:48.380 --> 00:02:49.810
So let's see.

00:02:49.810 --> 00:02:53.360
I could draw in that dimension.

00:02:53.360 --> 00:02:56.810
OK, a 9 there.

00:02:56.810 --> 00:02:59.010
Let me use my mouse
instead of my pen tool.

00:02:59.010 --> 00:03:00.500
It'll be easier.

00:03:00.500 --> 00:03:06.650
And then let me make
this 9, this side 9,

00:03:06.650 --> 00:03:09.970
and then I could
make this side 9.

00:03:09.970 --> 00:03:11.050
And then we are all set.

00:03:11.050 --> 00:03:11.680
It's a square.

00:03:11.680 --> 00:03:13.360
We see we have
four right angles.

00:03:13.360 --> 00:03:15.120
And now we can check our answer.

00:03:15.120 --> 00:03:17.330
And we got it right.