Derivative of inverse cosine

WEBVTT
Kind: captions
Language: en

00:00:00.966 --> 00:00:02.900
Voiceover: In the last video, we showed or

00:00:02.900 --> 00:00:05.433
we proved to ourselves that the

00:00:05.433 --> 00:00:07.366
derivative of the inverse sine of x

00:00:07.366 --> 00:00:09.700
is equal to 1 over the square root

00:00:09.700 --> 00:00:11.679
of 1 minus x squared.

00:00:11.679 --> 00:00:12.996
What I encourage you to do in this video

00:00:13.073 --> 00:00:15.500
is to pause it and try to do the same

00:00:15.500 --> 00:00:17.366
type of proof for the derivative

00:00:17.366 --> 00:00:20.366
of the inverse cosine of x.

00:00:20.366 --> 00:00:24.330
So, our goal here is to figure out ...

00:00:24.346 --> 00:00:25.940
I want the derivative with respect

00:00:25.940 --> 00:00:29.366
to x of the inverse cosine,

00:00:29.366 --> 00:00:31.038
inverse cosine of x.

00:00:31.038 --> 00:00:34.038
What is this going to be equal to?

00:00:34.038 --> 00:00:35.715
So, assuming you've had a go at it,

00:00:35.715 --> 00:00:37.259
let's work through it.

00:00:37.259 --> 00:00:38.966
So, just like last time, we could write,

00:00:38.966 --> 00:00:42.100
let's just set y being equal to this.

00:00:42.100 --> 00:00:44.812
y is equal the inverse cosine of x,

00:00:44.812 --> 00:00:46.925
which means the same thing is saying

00:00:46.925 --> 00:00:50.433
that x is equal to the cosine,

00:00:50.433 --> 00:00:51.782
cosine of y.

00:00:51.782 --> 00:00:52.704
I'll just take the derivative of

00:00:52.704 --> 00:00:55.500
both sides with respect to x.

00:00:55.500 --> 00:00:59.566
On the left hand side, you're
just going to have a 1.

00:00:59.566 --> 00:01:01.433
We're just going to have a 1.

00:01:01.433 --> 00:01:02.212
And on the right hand side,

00:01:02.319 --> 00:01:04.166
you're going to have the derivative of

00:01:04.166 --> 00:01:06.167
cosine y with respect to y, which

00:01:06.167 --> 00:01:10.300
is negative sine of y times the

00:01:10.300 --> 00:01:12.166
derivative of y with respect to x,

00:01:12.166 --> 00:01:16.700
which is dy, dx, and so we get ...

00:01:16.700 --> 00:01:18.300
Let's see if we divide both sides by

00:01:18.300 --> 00:01:24.433
negative sine of y, we
get dy, dx is equal to

00:01:24.433 --> 00:01:28.700
negative 1 over sine of y.

00:01:28.700 --> 00:01:30.766
Now, like we've seen
before, this is kind of

00:01:30.766 --> 00:01:34.300
satisfying, but we have our
derivative in terms of y.

00:01:34.300 --> 00:01:36.433
We want it in terms of x.

00:01:36.433 --> 00:01:38.633
And we know that x is cosine of y, so

00:01:38.633 --> 00:01:39.766
let's see if we can rewrite this

00:01:39.766 --> 00:01:41.633
bottom expression in terms of

00:01:41.633 --> 00:01:44.700
cosine of y instead of sine y.

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Well, we know when we saw in the

00:01:45.823 --> 00:01:49.167
last video from the pythagorean identity

00:01:49.167 --> 00:01:51.900
that cosine squared of y plus

00:01:51.900 --> 00:01:53.900
sine squared of y is equal to 1.

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We know that sine of y is equal to the

00:01:56.100 --> 00:02:00.100
square root of 1 minus
cosine squared of y.

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So, this is equal to negative 1.

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This is just a manipulation of the

00:02:04.433 --> 00:02:07.233
pythagorean trig identity.

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This is equal to 1 minus cosine.

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I can write like this,
cosine squared of y,

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but I'll write it like this

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because it'll make it
a little bit clearer.

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And what is cosine of y?

00:02:18.833 --> 00:02:21.033
Well, of course that is x, so this is

00:02:21.033 --> 00:02:28.500
equal to negative 1 over
the square root of 1 minus.

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Instead of writing cosine y ...

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Instead of writing cosine y ...

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I'm trying to switch colors.

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Instead of writing cosine y,

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we could write 1 minus x,

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1 minus x squared, so there you have it.

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The derivative with respect to x of the

00:02:44.566 --> 00:02:50.171
inverse cosine of x is ...
I think I lost that color,

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I'll do it in magenta ... is equal to

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negative 1 over the square
root of 1 minus x squared,

00:03:00.766 --> 00:03:03.233
so this is a neat thing.

00:03:03.233 --> 00:03:06.300
This right over here is
a neat thing to know.

00:03:06.300 --> 00:03:10.300
And of course, we should
compare it to the inverse,

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the derivative of the inverse sine.

00:03:11.612 --> 00:03:13.094
Actually, let me put them side by side,

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and we see that the only difference here

00:03:15.300 --> 00:03:19.766
is the sign, so let me
copy and paste that.

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I'll copy and paste it,

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I'm going to paste it down here,

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and now let's look at them side by side.

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So, we see for taking the derivative

00:03:28.966 --> 00:03:30.776
with respect to x of the inverse cosine

00:03:30.776 --> 00:03:33.300
function, we have a negative.

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A negative 1 over the square root of

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1 minus x squared.

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If we're looking at the derivative with

00:03:37.900 --> 00:03:39.433
respect to x of the inverse sine,

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it's the same expression
except now it is positive.


Derivative of inverse sine

WEBVTT
Kind: captions
Language: en

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What I would like to explore in this
video, is to see if

00:00:04.010 --> 00:00:08.480
we could figure out the derivative of Y is
with respect to X.

00:00:08.480 --> 00:00:15.090
If Y is equal to the inverse sine, the
inverse sine of X.

00:00:15.090 --> 00:00:17.280
And like always, I encourage you to pause
this

00:00:17.280 --> 00:00:19.250
video and try to figure this out on your
own.

00:00:19.250 --> 00:00:21.670
And I will give you two hints.

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First hint is, well, we don't know what
the derivative of sine inverse of x is,

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but we do know what the si-, what the
derivative of the sine of something is.

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And so if you, maybe if you rearrange this
and use

00:00:31.190 --> 00:00:36.360
some implicit differentiation, maybe you
can figure out what dy, dx is.

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Remember, this is right over here.

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This right over here is our goal.

00:00:41.100 --> 00:00:46.060
But, since you want to figure our the
derivative of this with respect to x.

00:00:46.060 --> 00:00:49.830
So, assuming you've had a go at it, so
let's work through this together.

00:00:49.830 --> 00:00:53.242
So, if y is the inverse sine of x, that's
just

00:00:53.242 --> 00:00:58.750
like saying that, that's equivalent to
saying that sine of y.

00:00:58.750 --> 00:01:03.140
Sine of Y is equal to X.

00:01:03.140 --> 00:01:05.580
Sine of Y is equal to X.

00:01:05.580 --> 00:01:08.070
So now we have things that we're a little
bit more familiar

00:01:08.070 --> 00:01:11.130
with, and now we can do a little bit of
implicit differentiation.

00:01:11.130 --> 00:01:14.665
We can take the derivative of both sides
with respect to X.

00:01:14.665 --> 00:01:18.550
So, derivative of the left-hand side with
respect to X

00:01:18.550 --> 00:01:23.570
and the derivative of the right-hand side
with respect to X.

00:01:23.570 --> 00:01:27.410
But what's the derivative of the left-hand
side with respect to X going to be?

00:01:27.410 --> 00:01:29.480
And here we just apply the chain rule.

00:01:29.480 --> 00:01:34.020
It's going to be the derivative of sine of
Y with respect to Y.

00:01:34.020 --> 00:01:37.808
Which is going to be, which is going to be
cosine

00:01:37.808 --> 00:01:41.950
of Y times the derivative of Y with
respect to X.

00:01:41.950 --> 00:01:49.140
So times dy, dx, times dy,dx.

00:01:49.140 --> 00:01:52.302
And the right-hand side, what's the
derivate of X with respect

00:01:52.302 --> 00:01:55.190
to X, well that's obviously just going to
be equal to one.

00:01:56.550 --> 00:02:01.670
And so we could solve for dy,dx, divide
both sides by cosine of Y.

00:02:01.670 --> 00:02:06.419
And we get the derivative of Y with
respect

00:02:06.419 --> 00:02:10.509
to X is equal to one over cosine of Y.

00:02:10.509 --> 00:02:14.490
Now this still isn't that satisfying cuz I
have the derivative in terms of Y.

00:02:14.490 --> 00:02:18.130
So let's see if we can re-express it in
terms of X.

00:02:18.130 --> 00:02:19.780
So, how could we do that?

00:02:19.780 --> 00:02:22.150
Well, we already know that X is equal to
sine of Y.

00:02:22.150 --> 00:02:22.950
Let me rewrite it.

00:02:22.950 --> 00:02:27.190
We already know that X is equal to sine of
Y.

00:02:27.190 --> 00:02:28.876
So, if we could rewrite this bottom

00:02:28.876 --> 00:02:31.080
expression in terms, instead of cosine of
Y.

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If we could use our trigonometric
identities

00:02:33.784 --> 00:02:35.633
to rewrite it in terms of sine of

00:02:35.633 --> 00:02:40.260
Y, then we'll be in good shape because X
is equal to sine of Y.

00:02:40.260 --> 00:02:42.010
Well, how can we do that?

00:02:42.010 --> 00:02:47.160
Well, we know from our trigonometric
identities, we know

00:02:47.160 --> 00:02:52.520
that sine squared of Y plus cosine squared
of Y is equal to one.

00:02:52.520 --> 00:02:54.664
Or, if we want to solve for cosine of

00:02:54.664 --> 00:02:58.060
Y, subtract sine squared of Y from both
sides.

00:02:58.060 --> 00:03:04.400
We know that cosine squared of Y is equal
to one minus sine squared of Y.

00:03:04.400 --> 00:03:10.154
Or that cosine of Y, just take the
principal root of both sides,

00:03:10.154 --> 00:03:16.570
is equal to the principal root of one
minus sine squared of Y.

00:03:16.570 --> 00:03:23.097
So, we could rewrite this as being equal
to one over, one over, instead

00:03:23.097 --> 00:03:29.700
of cosine of Y, we could rewrite it as one
minus sine squared of Y.

00:03:29.700 --> 00:03:31.080
Now why is this useful?

00:03:31.080 --> 00:03:33.340
Well, sine of Y is just X.

00:03:33.340 --> 00:03:35.840
So this is the same if we just substitute
back in, let

00:03:35.840 --> 00:03:38.220
me just write it that way so it's a little
bit clear.

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I could write it as sine Y squared.

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We know that this thing right over here is
X.

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So this is going to be equal to, and we
deserve a little bit of a drumroll.

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One over the square root of one minus,
instead of sin

00:03:51.710 --> 00:03:54.610
of Y, we know that X is equal to sin of Y.

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So, one minus X squared.

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And so, there you have it.

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The derivative with respect to X of the
inverse sine of X is equal to one over

00:04:04.929 --> 00:04:12.130
the square root of one minus X squared, so
let me just make that very clear.

00:04:12.130 --> 00:04:18.040
If you were to take the derivative with
respect to X of both sides of

00:04:18.040 --> 00:04:23.746
this, you get dy,dx is equal to this on
the right-hand side.

00:04:23.746 --> 00:04:29.892
Or we could say the derivative with
respect to X of the inverse sine

00:04:29.892 --> 00:04:36.260
of X is equal to one over the square root
of one minus X squared.

00:04:36.260 --> 00:04:39.450
Now you could always reprove this if your
memory starts to fail

00:04:39.450 --> 00:04:42.860
you, and actually, that is the best way to
really internalize this.

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But this is also just a good thing to
know, especially as we

00:04:45.433 --> 00:04:48.319
start, as we go into more and more
calculus and you see, you might

00:04:48.319 --> 00:04:51.521
see this in expression and you might say,
oh, okay, you know, that's

00:04:51.521 --> 00:04:54.750
the derivative of the inverse sine of X,
which might prove to be useful.


_-substitution: definite integral of exponential function

WEBVTT
Kind: captions
Language: en

00:00:00.566 --> 00:00:02.301
Sal: Let's see if we can
calculate the definite

00:00:02.301 --> 00:00:07.566
integral from zero to
one of x squared times

00:00:07.566 --> 00:00:13.500
two to the x to the third power d x.

00:00:13.500 --> 00:00:15.900
Like always I encourage
you to pause this video

00:00:15.900 --> 00:00:20.233
and see if you can figure
this out on your own.

00:00:20.940 --> 00:00:22.634
I'm assuming you've had a go at it.

00:00:22.634 --> 00:00:24.300
There's a couple of
interesting things here.

00:00:24.300 --> 00:00:26.100
The first thing, at least
that my brain does, it says,

00:00:26.100 --> 00:00:28.367
"I'm used to taking derivatives
and anti-derivatives

00:00:28.367 --> 00:00:31.700
of e to the x, not some
other base to the x."

00:00:31.700 --> 00:00:34.700
We know that the derivative
with respect to x of e to the x

00:00:34.700 --> 00:00:38.100
is e to the x, or we could
say that the anti-derivative

00:00:38.100 --> 00:00:43.700
of e to the x is equal
to e to the x plus c.

00:00:43.700 --> 00:00:46.900
Since I'm dealing with
something raised to,

00:00:47.930 --> 00:00:49.900
this particular situation,
something raised to

00:00:49.900 --> 00:00:52.500
a function of x, it seems
like I might want to put,

00:00:52.500 --> 00:00:56.433
I might want to change the base
here, but how do I do that?

00:00:56.433 --> 00:01:00.866
The way I would do that is
re-express two in terms of e.

00:01:01.100 --> 00:01:03.233
What would be two in terms of e?

00:01:03.233 --> 00:01:09.900
Two is equal to e, is
equal to e raised to the

00:01:09.900 --> 00:01:13.033
power that you need to
raise e to to get to two.

00:01:13.033 --> 00:01:14.367
What's the power that you have

00:01:14.367 --> 00:01:16.500
to raise two to to get to two?

00:01:16.500 --> 00:01:18.600
Well that's the natural log of two.

00:01:18.633 --> 00:01:21.633
Once again the natural
log of two is the exponent

00:01:21.633 --> 00:01:24.100
that you have to raise e to to get to two.

00:01:24.100 --> 00:01:27.700
If you actually raise e to
it you're going to get two.

00:01:27.700 --> 00:01:29.567
This is what two is.

00:01:29.567 --> 00:01:31.567
Now what is two to the x to the third?

00:01:31.567 --> 00:01:34.500
Well if we raise both sides
of this to the x to the third

00:01:34.500 --> 00:01:38.166
power, we raise both sides
to the x to the third power,

00:01:38.166 --> 00:01:40.967
two to the x to the third
is equal to, if I raise

00:01:40.967 --> 00:01:43.433
something to an exponent and
then raise that to an exponent,

00:01:43.433 --> 00:01:49.266
it's going to be equal to
e to the x to the third,

00:01:49.633 --> 00:01:53.000
x to the third, times
the natural log of two,

00:01:53.100 --> 00:01:56.134
times the natural log of two.

00:01:56.233 --> 00:01:59.166
That already seems pretty interesting.

00:01:59.166 --> 00:02:01.567
Let's rewrite this, and
actually what I'm going to do,

00:02:01.567 --> 00:02:04.100
let's just focus on the
indefinite integral first,

00:02:04.100 --> 00:02:05.166
see if we can figure that out.

00:02:05.166 --> 00:02:06.500
Then we can apply, then we can take,

00:02:06.500 --> 00:02:08.566
we can evaluate the definite ones.

00:02:08.566 --> 00:02:11.566
Let's just think about
this, let's think about

00:02:11.566 --> 00:02:15.033
the indefinite integral of x squared times

00:02:15.033 --> 00:02:18.500
two to the x to the third power d x.

00:02:18.500 --> 00:02:20.566
I really want to find the
anti-derivative of this.

00:02:20.566 --> 00:02:23.166
Well this is going to be
the exact same thing as

00:02:23.166 --> 00:02:28.433
the integral of, I'll
write my x squared still,

00:02:28.433 --> 00:02:30.366
but instead of two to the x to the third

00:02:30.366 --> 00:02:32.166
I'm going to write all of this business.

00:02:32.166 --> 00:02:33.833
Let me just copy and paste that.

00:02:33.833 --> 00:02:35.300
We already established
that this is the same

00:02:35.300 --> 00:02:38.500
thing as two to the x to the third power.

00:02:38.500 --> 00:02:42.867
Copy and paste, just like that.

00:02:43.433 --> 00:02:47.800
Then let me close it with a d x.

00:02:48.838 --> 00:02:51.767
I was able to get it in
terms of e as a base.

00:02:51.767 --> 00:02:53.434
That makes me a little
bit more comfortable

00:02:53.434 --> 00:02:55.366
but it still seems pretty complicated.

00:02:55.366 --> 00:02:57.700
You might be saying, "Okay, look.

00:02:57.700 --> 00:03:00.966
"Maybe u substitution
could be at play here."

00:03:00.966 --> 00:03:05.033
Because I have this crazy
expression, x to the third times

00:03:05.033 --> 00:03:08.000
the natural log of two, but
what's the derivative of that?

00:03:08.033 --> 00:03:09.366
Well that's going to be three x

00:03:09.366 --> 00:03:11.167
squared times the natural log of two,

00:03:11.167 --> 00:03:13.966
or three times the natural
log of two times x squared.

00:03:13.966 --> 00:03:16.166
That's just a constant times x squared.

00:03:16.166 --> 00:03:18.766
We already have a x squared
here so maybe we can engineer

00:03:18.766 --> 00:03:22.934
this a little bit to have
the constant there as well.

00:03:22.967 --> 00:03:24.233
Let's think about that.

00:03:24.233 --> 00:03:27.900
If we made this, if we defined this as u,

00:03:27.900 --> 00:03:33.233
if we said u is equal
to x to the third times

00:03:33.233 --> 00:03:36.234
the natural log of two,
what is du going to be?

00:03:36.234 --> 00:03:39.900
du is going to be, it's
going to be, well natural

00:03:39.900 --> 00:03:42.033
log of two is just a
constant so it's going to be

00:03:42.033 --> 00:03:45.933
three x squared times
the natural log of two.

00:03:46.033 --> 00:03:47.433
We could actually just change the order

00:03:47.433 --> 00:03:49.033
we're multiplying a little bit.

00:03:49.033 --> 00:03:50.500
We could say that this
is the same thing as

00:03:50.500 --> 00:03:55.567
x squared times three natural log of two,

00:03:55.567 --> 00:03:58.566
which is the same thing just
using logarithm properties,

00:03:58.566 --> 00:04:03.766
as x squared times the natural
log of two to the third power.

00:04:03.766 --> 00:04:05.100
Three natural log of two is the same thing

00:04:05.100 --> 00:04:07.166
as the natural log of
two to the third power.

00:04:07.166 --> 00:04:13.333
This is equal to x squared
times the natural log of eight.

00:04:13.700 --> 00:04:16.433
Let's see, if this is u, where is du?

00:04:16.433 --> 00:04:19.466
Oh, and of course we can't forget the dx.

00:04:19.966 --> 00:04:25.800
This is a dx right over here, dx, dx, dx.

00:04:25.833 --> 00:04:29.767
Where is the du? Well we have
a dx. Let me circle things.

00:04:29.767 --> 00:04:32.767
You have a dx here, you have a dx there.

00:04:32.767 --> 00:04:36.100
You have an x squared here,
you have an x squared here.

00:04:36.100 --> 00:04:38.234
So really all we need is,

00:04:38.234 --> 00:04:40.900
all we need here is the
natural log of eight.

00:04:40.900 --> 00:04:43.834
Ideally we would have the
natural log of eight right over

00:04:43.834 --> 00:04:46.966
here, and we could put it
there as long as we also,

00:04:46.966 --> 00:04:49.233
we could multiply by
the natural log of eight

00:04:49.233 --> 00:04:53.333
as long as we also divide
by a natural log of eight.

00:04:53.366 --> 00:04:56.300
We can do it like right over here,

00:04:56.300 --> 00:04:58.833
we could divide by natural log of eight.

00:04:58.833 --> 00:05:01.366
But we know that the
anti-derivative of some constant

00:05:01.366 --> 00:05:04.033
times a function is the
same thing as a constant

00:05:04.033 --> 00:05:06.300
times the anti-derivative
of that function.

00:05:06.300 --> 00:05:08.433
We could just take that on the outside.

00:05:08.433 --> 00:05:12.333
It's one over the natural log of eight.

00:05:12.566 --> 00:05:15.433
Let's write this in terms of u and du.

00:05:15.433 --> 00:05:18.566
This simplifies to one over the natural

00:05:18.566 --> 00:05:23.346
log of eight times the anti-derivative of

00:05:24.453 --> 00:05:31.766
e to the u, e to the u, that's the u, du.

00:05:31.766 --> 00:05:36.509
This times this times that is du, du.

00:05:36.509 --> 00:05:38.700
And this is straightforward,

00:05:38.700 --> 00:05:40.900
we know what this is going to be.

00:05:40.900 --> 00:05:43.100
This is going to be equal
to, let me just write

00:05:43.100 --> 00:05:45.600
the one over natural
log of eight out here,

00:05:45.633 --> 00:05:53.879
one over natural log of
eight times e to the u,

00:05:55.771 --> 00:05:57.700
and of course if we're
thinking in terms of just

00:05:57.700 --> 00:06:00.433
anti-derivative there would
be some constant out there.

00:06:00.433 --> 00:06:02.900
Then we would just
reverse the substitution.

00:06:02.900 --> 00:06:04.500
We already know what u is.

00:06:04.500 --> 00:06:07.300
This is going to be equal
to, the anti-derivative of

00:06:07.300 --> 00:06:11.566
this expression is one over
the natural log of eight

00:06:11.566 --> 00:06:15.367
times e to the, instead
of u, we know that u is

00:06:15.367 --> 00:06:19.100
x to the third times
the natural log of two.

00:06:19.100 --> 00:06:21.866
And of course we could put a plus c there.

00:06:22.100 --> 00:06:24.234
Now, going back to the original problem.

00:06:24.234 --> 00:06:26.566
We just need to evaluate
the anti-derivative

00:06:26.566 --> 00:06:29.500
of this at each of these points.

00:06:29.500 --> 00:06:30.967
Let's rewrite that.

00:06:30.967 --> 00:06:36.066
Given what we just figured out,
let me copy and paste that.

00:06:36.301 --> 00:06:39.120
This is just going to be equal to,

00:06:40.181 --> 00:06:43.966
it's going to be equal to
the anti-derivative evaluated

00:06:43.966 --> 00:06:47.100
at one minus the anti-derivative
evaluated at zero.

00:06:47.100 --> 00:06:48.033
We don't have to worry about the

00:06:48.033 --> 00:06:50.033
constants because those will cancel out.

00:06:50.033 --> 00:06:53.566
So we are going to get,
we are going to get one--

00:06:53.566 --> 00:06:55.966
Let me evaluate it first at one.

00:06:56.596 --> 00:06:59.500
You're going to get one
over the natural log of

00:06:59.500 --> 00:07:05.433
eight times e to the
one to the third power,

00:07:05.433 --> 00:07:08.100
which is just one, times
the natural log of two,

00:07:08.100 --> 00:07:10.834
natural log of two,
that's evaluated at one.

00:07:10.834 --> 00:07:15.066
Then we're going to have
minus it evaluated it at zero.

00:07:15.100 --> 00:07:18.033
It's going to be one
over the natural log of

00:07:18.033 --> 00:07:21.633
eight times e to the, well when x is zero

00:07:21.633 --> 00:07:23.766
this whole thing is going to be zero.

00:07:23.766 --> 00:07:29.100
Well e to the zero is just
one, and e to the natural

00:07:29.100 --> 00:07:32.233
log of two, well that's
just going to be two,

00:07:32.233 --> 00:07:33.966
we already established that early on,

00:07:33.966 --> 00:07:35.766
this is just going to be equal to two.

00:07:35.766 --> 00:07:39.366
We are left with two over the
natural log of eight minus

00:07:39.366 --> 00:07:42.500
one over the natural log of
eight, which is just going

00:07:42.500 --> 00:07:47.533
to be equal to one over
the natural log of eight.

00:07:47.900 --> 00:07:52.266
And we are, and we are done.


16 Possible values for t

WEBVTT
Kind: captions
Language: en

00:00:00.674 --> 00:00:02.521
- [Voiceover] Let inverted, or upside down

00:00:02.521 --> 00:00:05.909
triangle x, be defined
as x plus one over x.

00:00:05.909 --> 00:00:07.012
Let me write that down.

00:00:07.012 --> 00:00:08.696
Inverted triangle x,

00:00:08.696 --> 00:00:12.493
is equal to x plus one over x.

00:00:12.493 --> 00:00:16.417
For all non-zero integers x,

00:00:16.417 --> 00:00:19.237
all right, if inverted or upside down

00:00:19.237 --> 00:00:21.152
triangle x is equal to t.

00:00:21.152 --> 00:00:24.634
So if this whole thing is equal to t,

00:00:24.634 --> 00:00:26.470
where t is an integer,

00:00:26.470 --> 00:00:30.498
which of the following
is a possible value of t.

00:00:30.498 --> 00:00:32.158
So let's think about this a little bit.

00:00:32.158 --> 00:00:34.758
So, what you could try to do,

00:00:34.758 --> 00:00:37.277
is you could take each of these choices,

00:00:37.277 --> 00:00:39.355
and substitute it in for t,

00:00:39.355 --> 00:00:41.549
and then try to solve for x,

00:00:41.549 --> 00:00:44.788
and see if you can find
x's that are integers.

00:00:44.788 --> 00:00:47.157
But instead of doing
each of them one by one,

00:00:47.157 --> 00:00:50.210
let's just rearrange this equation

00:00:50.210 --> 00:00:53.610
where we can solve for
x as a function of t.

00:00:53.610 --> 00:00:55.713
And then we could try
the different t values

00:00:55.713 --> 00:00:59.230
out (mumbles), and say,
can we get a non-zero

00:00:59.230 --> 00:01:02.262
integer value for x?

00:01:02.262 --> 00:01:04.468
So if we take this, and if we multiply

00:01:04.468 --> 00:01:07.241
both sides by x, we're going to get

00:01:07.241 --> 00:01:09.946
x times x is x squared.

00:01:09.946 --> 00:01:12.664
One over x times x is one.

00:01:12.664 --> 00:01:16.598
And then t times x is tx.

00:01:16.598 --> 00:01:18.503
Now, it looks like we have a quadratic,

00:01:18.503 --> 00:01:20.697
we could try to get into a standard form

00:01:20.697 --> 00:01:22.752
so we could try to factor it.

00:01:22.752 --> 00:01:25.631
So let's subtract tx from both sides.

00:01:25.631 --> 00:01:29.125
And so if we subtract tx,

00:01:29.125 --> 00:01:31.749
from both sides, we are going

00:01:31.749 --> 00:01:34.686
to be left with, on the left-hand side,

00:01:34.686 --> 00:01:39.584
x squared minus tx plus one,

00:01:39.584 --> 00:01:43.125
and that's going to be equal to zero.

00:01:43.125 --> 00:01:45.145
Now this is interesting right over here.

00:01:45.145 --> 00:01:47.932
We have, kind of, a traditional quadratic.

00:01:47.932 --> 00:01:50.648
A traditional quadratic
equation right over here.

00:01:50.648 --> 00:01:52.410
We just have to think about, well,

00:01:52.410 --> 00:01:55.365
for what values of t can
we actually factor this

00:01:55.365 --> 00:01:59.148
so that we can integer solutions for x.

00:01:59.148 --> 00:02:00.101
So, actually, let's just write down

00:02:00.101 --> 00:02:01.634
all the options over here.

00:02:01.634 --> 00:02:04.605
So if t is equal to one,

00:02:04.605 --> 00:02:06.787
then this thing is going to be x squared

00:02:06.787 --> 00:02:08.172
minus one x, so we could just say,

00:02:08.172 --> 00:02:11.081
x squared minus x plus one.

00:02:11.081 --> 00:02:13.402
If t is equal to zero,

00:02:13.402 --> 00:02:14.872
then this middle term disappears,

00:02:14.872 --> 00:02:19.451
and you're just left
with x squared plus one.

00:02:19.451 --> 00:02:23.374
If t is equal to negative one,

00:02:23.374 --> 00:02:26.172
then this thing becomes x squared,

00:02:26.172 --> 00:02:29.724
negative, negative one,
so it's plus x plus one.

00:02:29.724 --> 00:02:32.626
If t is equal to negative two,

00:02:32.626 --> 00:02:37.626
then you get x squared
plus two x, plus one.

00:02:37.631 --> 00:02:40.344
And then if t is equal to...

00:02:40.344 --> 00:02:41.786
I'm running out of colors.

00:02:41.786 --> 00:02:43.638
If t is equal to negative three,

00:02:43.638 --> 00:02:47.794
you have x squared plus
negative, negative three,

00:02:47.794 --> 00:02:50.558
so that's plus three x plus one.

00:02:50.558 --> 00:02:52.834
So which of these can you factor?

00:02:52.834 --> 00:02:54.274
Well, it might jump out at you that,

00:02:54.274 --> 00:02:55.980
hey, this is a perfect
square right over here,

00:02:55.980 --> 00:02:57.246
I could factor this one.

00:02:57.246 --> 00:03:02.246
This is x plus one times x plus one.

00:03:02.400 --> 00:03:05.778
One plus one is two,
and one times one is one

00:03:05.778 --> 00:03:07.066
right over here.

00:03:07.066 --> 00:03:09.145
So because I can factor this,

00:03:09.145 --> 00:03:12.140
I know that this right over here

00:03:12.140 --> 00:03:14.242
is going to be my answer.

00:03:14.242 --> 00:03:19.025
If t is equal to negative two,

00:03:19.025 --> 00:03:21.501
then all of this stuff boils down to that,

00:03:21.501 --> 00:03:24.530
and we could say that x squared

00:03:24.530 --> 00:03:27.399
minus t, which is plus two,

00:03:27.399 --> 00:03:31.056
plus two x plus one is equal to zero.

00:03:31.056 --> 00:03:34.015
The left-hand side is
just x plus one squared

00:03:34.015 --> 00:03:37.139
is equal to zero, which
means that x plus one

00:03:37.139 --> 00:03:40.458
is equal to zero, which means that

00:03:40.458 --> 00:03:43.639
x would be equal to negative one.

00:03:43.639 --> 00:03:48.456
So when t is equal to negative two,

00:03:48.456 --> 00:03:50.895
that means that x is
equal to negative one,

00:03:50.895 --> 00:03:54.377
and negative one is a non-zero integer.

00:03:54.377 --> 00:03:55.457
You can look at these other ones,

00:03:55.457 --> 00:03:56.711
and you say, well, yeah, this is very hard

00:03:56.711 --> 00:03:58.314
to factor, I need to take the product

00:03:58.314 --> 00:04:00.298
of two numbers, whose positive one,

00:04:00.298 --> 00:04:02.073
and their sum is negative one.

00:04:02.073 --> 00:04:06.750
Well I can't think of two integers

00:04:06.750 --> 00:04:08.145
that would do that.

00:04:08.145 --> 00:04:10.466
Same thing here, x squared plus one,

00:04:10.466 --> 00:04:12.568
how do I factor that?

00:04:12.568 --> 00:04:13.948
There's no obvious way, at least

00:04:13.948 --> 00:04:15.551
in my brain, of factoring that.

00:04:15.551 --> 00:04:17.187
Same thing over here, two numbers,

00:04:17.187 --> 00:04:19.406
whose product is one and whose sum

00:04:19.406 --> 00:04:21.031
is positive one, well, I needed

00:04:21.031 --> 00:04:23.305
two integers, and I
can't think of any there.

00:04:23.305 --> 00:04:24.269
And same thing over here,

00:04:24.269 --> 00:04:25.743
two numbers whose product is one,

00:04:25.743 --> 00:04:27.683
and whose sum is three,

00:04:27.683 --> 00:04:28.599
can't think of it.

00:04:28.599 --> 00:04:32.277
But this one, right over here, you can.


13 Possible values for x

WEBVTT
Kind: captions
Language: en

00:00:00.743 --> 00:00:02.831
- [Voiceover] If x and y are integers,

00:00:02.831 --> 00:00:05.854
seven is less than y is less than 16,

00:00:05.854 --> 00:00:08.131
and x over y is equal to 2/5,

00:00:08.486 --> 00:00:12.322
how many possible values are there for x?

00:00:12.774 --> 00:00:15.147
So x and y have to be integers,

00:00:15.558 --> 00:00:18.445
and y is greater than
seven and less than 16.

00:00:18.450 --> 00:00:20.537
So let's, before we even
think about this constraint

00:00:20.537 --> 00:00:22.040
right over here and this constraint,

00:00:23.161 --> 00:00:25.501
if we just had this
constraint right over here,

00:00:25.501 --> 00:00:27.189
what are the possible values for y?

00:00:27.189 --> 00:00:29.686
Well, y is an integer greater than seven,

00:00:29.686 --> 00:00:34.157
so let's see, it would
be eight, nine, 10, 11,

00:00:34.572 --> 00:00:38.894
12, 13, 14, 15,

00:00:39.067 --> 00:00:41.979
and it can't be equal to 16
so we'll stop right over here.

00:00:41.979 --> 00:00:44.544
So just with that constraint,
here are all our candidate ys.

00:00:44.950 --> 00:00:48.286
Now they're saying x
over y is equal to 2/5,

00:00:48.286 --> 00:00:50.229
or another, let me just rewrite that.

00:00:50.229 --> 00:00:53.385
So they're saying x
over y is equal to 2/5,

00:00:53.799 --> 00:00:55.925
or if you multiply both sides by y,

00:00:55.925 --> 00:00:59.998
you get x is equal to 2/5y.

00:01:00.792 --> 00:01:02.784
Now, if x is equal to 2/5y,

00:01:02.784 --> 00:01:06.221
so over here you would
have eight times 2/5.

00:01:06.568 --> 00:01:07.873
So let me write this down.

00:01:07.873 --> 00:01:10.422
So these are our possible ys,

00:01:12.275 --> 00:01:14.065
and then our possible xs,

00:01:14.065 --> 00:01:16.049
eight times 2/5.

00:01:16.692 --> 00:01:18.877
Now is this thing right over
here going to be an integer?

00:01:18.877 --> 00:01:21.068
Remember, x and y need to be integers.

00:01:21.068 --> 00:01:23.801
Well no, eight isn't divisible by five.

00:01:23.932 --> 00:01:26.784
Eight times two is 16, 16
isn't divisible by five.

00:01:27.157 --> 00:01:29.260
And so this is not going to be an integer.

00:01:29.644 --> 00:01:33.198
Is nine times 2/5, is that
going to be an integer?

00:01:33.545 --> 00:01:35.462
No, nine isn't divisible by five.

00:01:35.618 --> 00:01:37.181
I think you start to see a pattern here.

00:01:37.533 --> 00:01:40.578
10 times 2/5, is that
going to be an integer?

00:01:40.578 --> 00:01:41.653
Sure!

00:01:41.666 --> 00:01:43.081
10 is divisible by five.

00:01:43.081 --> 00:01:44.745
10 divided by five is two,

00:01:44.745 --> 00:01:47.601
10 divided by five is two, times two,

00:01:47.601 --> 00:01:49.509
this is equal to four.

00:01:49.843 --> 00:01:52.908
So this is a possible value for x,

00:01:52.969 --> 00:01:54.795
and frankly this is a possible value,

00:01:55.471 --> 00:01:58.039
this is a possible value for y,

00:01:58.039 --> 00:02:00.535
if we assume that they
both have to be integers.

00:02:01.003 --> 00:02:02.352
And so we don't have to test all of these,

00:02:02.352 --> 00:02:03.103
we just say,

00:02:03.103 --> 00:02:05.444
"Look, which of these are
also divisible by five?"

00:02:05.744 --> 00:02:07.360
Well, 10 is divisible by five,

00:02:07.360 --> 00:02:09.540
and 15 is divisible by five.

00:02:09.615 --> 00:02:12.796
None of these other ones
are divisible by five.

00:02:12.959 --> 00:02:14.192
So if y is 15,

00:02:14.192 --> 00:02:16.103
we don't even have to
know what the value is,

00:02:16.103 --> 00:02:19.947
we could say x is going
to be 15 times 2/5,

00:02:20.319 --> 00:02:22.579
which is 15 divided by five is three,

00:02:22.719 --> 00:02:24.375
times two which is six.

00:02:24.375 --> 00:02:27.017
So there are two possible values,

00:02:27.017 --> 00:02:30.236
there are two possible values for x.


4 Polygon perimeter

WEBVTT
Kind: captions
Language: en

00:00:00.416 --> 00:00:02.230
- [Voiceover] In the figure above,

00:00:02.230 --> 00:00:05.337
CDE is an equilateral triangle.

00:00:05.337 --> 00:00:07.106
So that means this side
is equal to that side

00:00:07.106 --> 00:00:08.465
is equal to that side.

00:00:08.465 --> 00:00:12.484
AB and ABCE is a square with an area of 1.

00:00:12.617 --> 00:00:14.413
So this is a square with an area of 1.

00:00:14.737 --> 00:00:18.684
What is the perimeter of polygon ABCDE?

00:00:20.861 --> 00:00:23.796
Well, since we know this
is a square with area 1,

00:00:23.796 --> 00:00:25.429
we know that this

00:00:26.736 --> 00:00:29.032
is equivalent to that right over there,

00:00:29.032 --> 00:00:31.665
and the product of this
number is going to be 1.

00:00:31.665 --> 00:00:32.918
So it has to be 1.

00:00:33.131 --> 00:00:34.558
This times this is 1.

00:00:34.730 --> 00:00:37.332
They're the same thing, so
that is 1, and that is 1.

00:00:37.636 --> 00:00:40.305
Well, it's a square, so that
side is also going to be 1.

00:00:40.652 --> 00:00:42.360
And this side is also going to be 1.

00:00:42.916 --> 00:00:45.384
Now, we were just told that
this is an equilateral triangle.

00:00:45.548 --> 00:00:47.628
The length of CD is the
same as DE is the same

00:00:47.628 --> 00:00:49.496
as the length of CE.

00:00:49.496 --> 00:00:51.272
Well CE has the length 1,

00:00:51.272 --> 00:00:53.987
so that has length 1,
and that has length 1.

00:00:54.488 --> 00:00:56.519
So what's the perimeter
of the whole polygon?

00:00:56.519 --> 00:00:58.240
Well, if we start at A let's just see,

00:00:58.240 --> 00:01:00.939
one, two, three, four, five.

00:01:01.105 --> 00:01:02.453
It's a five-sided polygon,

00:01:02.453 --> 00:01:04.221
all of the sides have the same length,

00:01:04.221 --> 00:01:05.981
they're all exactly have a length of 1,

00:01:05.981 --> 00:01:09.425
so the perimeter is equal to 5.


10 Rope cutting

WEBVTT
Kind: captions
Language: en

00:00:00.657 --> 00:00:02.572
- A student was given a piece of rope

00:00:02.572 --> 00:00:04.731
and told to cut it into two equal pieces,

00:00:04.731 --> 00:00:06.867
keep one piece, and pass the other piece

00:00:06.867 --> 00:00:07.947
to the next student.

00:00:07.947 --> 00:00:10.432
Each student was to repeat this process

00:00:10.432 --> 00:00:11.732
until every student in the class

00:00:11.732 --> 00:00:13.972
had exactly one piece of rope.

00:00:13.972 --> 00:00:16.155
Which of the following
could be the fraction

00:00:16.155 --> 00:00:19.150
of the original rope that
one of the students had?

00:00:19.150 --> 00:00:21.797
So the key realization
here is that each student

00:00:21.797 --> 00:00:22.829
is going to have--

00:00:22.829 --> 00:00:24.351
The first student's going
to have half the rope,

00:00:24.351 --> 00:00:25.408
and then each student after that's

00:00:25.408 --> 00:00:26.986
going to have half of that.

00:00:26.986 --> 00:00:28.750
So all of the possible fractions

00:00:28.750 --> 00:00:30.329
are going to be powers of 2.

00:00:30.329 --> 00:00:32.929
So one-half, one-half times 2,

00:00:32.929 --> 00:00:34.811
one-half times one-half is one-fourth,

00:00:34.811 --> 00:00:36.993
one-half to the third power is one-eighth,

00:00:36.993 --> 00:00:40.209
one-half to the fourth
power is one-sixteenth.

00:00:40.209 --> 00:00:41.334
And we could keep going,

00:00:41.334 --> 00:00:44.272
but lucky for us we
already see the choice.

00:00:44.272 --> 00:00:45.746
One-sixteenth.

00:00:45.746 --> 00:00:47.488
Now, just to make that
a little bit clearer,

00:00:47.488 --> 00:00:49.043
let's work it out a little bit more.

00:00:49.043 --> 00:00:50.251
Just in case that seemed a little bit

00:00:50.251 --> 00:00:52.585
too hand-wavey for you.

00:00:52.585 --> 00:00:55.452
So the first student,
you have the whole rope,

00:00:55.452 --> 00:00:59.166
cuts it in half and keeps half of it,

00:00:59.166 --> 00:01:00.351
and hands that half--

00:01:00.351 --> 00:01:02.592
So, they get to keep half,

00:01:02.592 --> 00:01:04.507
and then hands the remaining half

00:01:04.507 --> 00:01:05.343
to the next student.

00:01:05.343 --> 00:01:08.443
So the next student has that half,

00:01:08.443 --> 00:01:10.589
but then has to cut it in half.

00:01:10.589 --> 00:01:13.143
And so that student is
going to be able to keep

00:01:13.143 --> 00:01:15.222
what I just circled, one-fourth,

00:01:15.222 --> 00:01:17.382
and then that's half of a half,

00:01:17.382 --> 00:01:18.473
which is one-fourth.

00:01:18.473 --> 00:01:19.970
And then hand that one-fourth over

00:01:19.970 --> 00:01:21.271
to the next student.

00:01:21.271 --> 00:01:23.349
The next student is going
to keep half of that,

00:01:23.349 --> 00:01:24.311
which I just circled,

00:01:24.311 --> 00:01:25.485
which is one-eighth.

00:01:25.485 --> 00:01:26.286
And I think you get--

00:01:26.286 --> 00:01:28.688
You see the general idea here.

00:01:28.688 --> 00:01:32.507
This is one-half, this is one-fourth,

00:01:32.810 --> 00:01:34.482
this is one-eighth.

00:01:34.482 --> 00:01:35.886
One-half to the first power,

00:01:35.886 --> 00:01:37.140
one-half to the second power,

00:01:37.140 --> 00:01:38.371
one-half to the third power,

00:01:38.371 --> 00:01:40.031
and you could keep on going.

00:01:40.031 --> 00:01:42.388
And of course, one-half
to the fourth power

00:01:42.388 --> 00:01:43.989
if we go one more step.

00:01:43.989 --> 00:01:45.325
We go one more step,

00:01:45.325 --> 00:01:47.066
so now you have one-eighth of a rope

00:01:47.066 --> 00:01:49.191
and the student cuts it in half

00:01:49.191 --> 00:01:51.515
and keeps half of it.

00:01:51.515 --> 00:01:53.975
That's one-sixteenth right over there.

00:01:53.975 --> 00:01:55.671
Which is what we actually picked.


11 Function equaling zero

WEBVTT
Kind: captions
Language: en

00:00:00.755 --> 00:00:02.856
- [Sal] Which of the
following is the graph of a

00:00:02.856 --> 00:00:07.344
function F such that F
of X is equal to zero

00:00:07.344 --> 00:00:12.312
for exactly two values of X
between negative five and five?

00:00:12.312 --> 00:00:15.192
So let's see, in all
of them they've plotted

00:00:15.192 --> 00:00:18.716
where X equals negative
five and X equals five.

00:00:18.716 --> 00:00:22.002
F of X needs to be equal
to zero for two values of X

00:00:22.002 --> 00:00:24.544
between negative five and five.

00:00:24.544 --> 00:00:27.503
So this function right over
here, this never gets to zero.

00:00:27.503 --> 00:00:29.861
In order to get to zero
it needs to intersect,

00:00:29.861 --> 00:00:32.821
it needs to touch on the
x-axis at some point.

00:00:32.821 --> 00:00:35.224
This one stays positive the whole time.

00:00:35.224 --> 00:00:38.254
Notice all the values for
this X right over here,

00:00:38.254 --> 00:00:40.472
F of X, it's a positive value.

00:00:40.472 --> 00:00:42.422
For this value of X right over here,

00:00:42.422 --> 00:00:46.060
F of X is a positive value,
so rule that one out.

00:00:46.060 --> 00:00:51.017
This one equals zero here, here and here.

00:00:51.017 --> 00:00:55.138
So it equals zero three
times, for exactly three

00:00:55.138 --> 00:00:58.017
values of X between
negative five and five,

00:00:58.017 --> 00:00:59.828
but they said exactly two values,

00:00:59.828 --> 00:01:01.651
so we'll rule that one out.

00:01:01.651 --> 00:01:05.598
This one is right at
positive five, you could say

00:01:05.598 --> 00:01:08.145
it's at one value if we
include the two boundary points

00:01:08.145 --> 00:01:10.504
right over here, but we said exactly two.

00:01:10.504 --> 00:01:12.408
This is kind of the other
side of it where we're

00:01:12.408 --> 00:01:15.984
only at negative five,
we need to be two times,

00:01:15.984 --> 00:01:17.586
so rule that one out.

00:01:17.586 --> 00:01:21.649
This one, we intersect the
x-axis exactly two times.

00:01:21.649 --> 00:01:24.563
F of X is equal to zero
for this value of X,

00:01:24.563 --> 00:01:27.570
F of X is equal to zero and F
of X is zero right over there.

00:01:27.570 --> 00:01:28.847
So that's...

00:01:28.847 --> 00:01:30.600
This is our choice.

00:01:30.600 --> 00:01:32.202
Another thing that could
have looked like that,

00:01:32.202 --> 00:01:33.804
maybe if it looked something like that.

00:01:33.804 --> 00:01:36.288
We would've had F of X
equal zero there and there.

00:01:36.288 --> 00:01:39.149
Either way, we pick D.


26 Inconsistent science

WEBVTT
Kind: captions
Language: en

00:00:00.075 --> 00:00:02.654
"Most of the hypotheses that Kepler

00:00:02.654 --> 00:00:05.723
developed to explain physical forces were

00:00:05.723 --> 00:00:09.637
later rejected as inconsistent to Newtonian

00:00:09.637 --> 00:00:13.135
theory, [or we have] No error." So, the

00:00:13.135 --> 00:00:14.664
part that seemed a little strange to me

00:00:14.664 --> 00:00:16.869
is when they talk about, "inconsistent to".

00:00:16.869 --> 00:00:18.513
You don't say that you are "inconsistent

00:00:18.513 --> 00:00:19.926
to" something, you say that you are

00:00:19.926 --> 00:00:24.689
"inconsistent [with]" something. Study A

00:00:24.689 --> 00:00:26.337
is not "inconsistent to" Study B, you would

00:00:26.337 --> 00:00:28.484
say, Study A is "inconsistent [with]"

00:00:28.484 --> 00:00:31.095
Study B. So, this should be "inconsistent

00:00:31.095 --> 00:00:32.703
[with]", and if I were to rewrite this,

00:00:32.703 --> 00:00:34.479
I would write this as: "Most of the

00:00:34.479 --> 00:00:37.219
hypotheses that Kepler developed to

00:00:37.219 --> 00:00:40.132
explain physical forces were later rejected

00:00:40.132 --> 00:00:45.955
as [being] inconsistent [with] Newtonian

00:00:45.955 --> 00:00:48.638
theory". That sounds a lot better,

00:00:48.638 --> 00:00:52.929
"inconsistent [with]". So, the error is in D.


7 Maximum distance on two spheres

WEBVTT
Kind: captions
Language: en

00:00:00.894 --> 00:00:03.471
- [Voiceover] Two spheres,
one with radius seven,

00:00:04.686 --> 00:00:06.629
and one with radius seven,

00:00:09.119 --> 00:00:13.362
and one with radius four
are tangent to each other.

00:00:13.362 --> 00:00:15.684
If P is any point on one sphere

00:00:15.684 --> 00:00:17.704
and Q is any point on the other sphere,

00:00:17.704 --> 00:00:20.409
what is the maximum possible length of PQ?

00:00:20.409 --> 00:00:21.941
So, let's think about this.

00:00:21.941 --> 00:00:24.809
So, let me draw the
sphere with radius seven.

00:00:24.809 --> 00:00:25.668
And obviously the sphere is

00:00:25.668 --> 00:00:26.806
a three dimensional object.

00:00:26.806 --> 00:00:27.827
And I'm going to try and draw it

00:00:27.827 --> 00:00:29.209
in two dimensions.

00:00:29.209 --> 00:00:31.066
So, a circle you can kind of use

00:00:31.066 --> 00:00:33.469
a cross section of it.

00:00:33.469 --> 00:00:35.861
So, this right over here,

00:00:35.861 --> 00:00:38.960
cross section at its
widest point of the sphere.

00:00:38.960 --> 00:00:41.061
So, it has radius seven.

00:00:41.061 --> 00:00:42.849
So, it has radius seven

00:00:42.849 --> 00:00:46.100
and it is tangent to another sphere

00:00:46.100 --> 00:00:48.166
that has radius four.

00:00:48.166 --> 00:00:51.568
So, it has radius, it has radius four.

00:00:51.568 --> 00:00:52.595
So, the other one might look something

00:00:52.595 --> 00:00:53.609
like this, once again.

00:00:53.609 --> 00:00:55.306
It's a cross section.

00:00:55.306 --> 00:00:57.302
It's tangent at exactly this point.

00:00:57.302 --> 00:00:58.568
And I've just drawn circles,

00:00:58.568 --> 00:01:01.528
but then you can imagine
these are actually spheres.

00:01:01.528 --> 00:01:04.221
I'll try to give it some actual depth,

00:01:04.221 --> 00:01:06.218
so that you recognize
that these are not just,

00:01:06.218 --> 00:01:09.898
you can imagine balloons
or balls of this shape.

00:01:09.898 --> 00:01:13.079
Now, P is any point on one sphere

00:01:13.079 --> 00:01:15.006
and Q is any point on the other sphere.

00:01:15.006 --> 00:01:17.386
And we want the maximum length of PQ.

00:01:17.386 --> 00:01:18.779
So, we want to get these points P and Q

00:01:18.779 --> 00:01:21.600
as far away as possible from each other.

00:01:21.600 --> 00:01:24.003
So, if I was just
dealing with two circles,

00:01:24.003 --> 00:01:27.056
I'd stick P right over
here, all the way here.

00:01:27.056 --> 00:01:29.146
And I would stick Q,

00:01:29.146 --> 00:01:32.420
I would stick Q all the way over there

00:01:32.420 --> 00:01:34.161
on the other circle.

00:01:34.161 --> 00:01:36.297
And so the distance
between them is going to be

00:01:36.297 --> 00:01:39.664
essentially the sum of the two diameters.

00:01:39.664 --> 00:01:41.335
So, it's going to be,

00:01:41.335 --> 00:01:44.435
it's going to be 14 plus,

00:01:44.435 --> 00:01:49.102
14 plus eight and 14 plus eight,

00:01:49.102 --> 00:01:54.102
14 plus eight is going to be equal to 22.

00:01:54.221 --> 00:01:55.998
And so, if I can do that with two circles

00:01:55.998 --> 00:01:57.913
of these radii, I could obviously do these

00:01:57.913 --> 00:01:58.784
as spheres as well.

00:01:58.784 --> 00:02:00.281
You could imagine these
are the cross sections

00:02:00.281 --> 00:02:01.419
of those spheres.

00:02:01.419 --> 00:02:03.323
So, I can get to a distance of 22,

00:02:03.323 --> 00:02:05.738
but am I sure that that's
the furthest distance?

00:02:05.738 --> 00:02:08.756
Well, you could reason
through it that look

00:02:08.756 --> 00:02:11.872
if you were to try to
bring P out a little bit

00:02:11.887 --> 00:02:15.176
more unto the surface
here, on the surface,

00:02:15.176 --> 00:02:17.149
yeah there's going to be
a little bit of distance

00:02:17.149 --> 00:02:19.541
now that it's kind of
going out of the page,

00:02:19.541 --> 00:02:22.060
if you're going out of
the surface right here.

00:02:22.060 --> 00:02:23.500
But, it's getting closer,

00:02:23.500 --> 00:02:26.239
essentially, along the page.

00:02:26.239 --> 00:02:28.166
So, maybe those two things trade off,

00:02:28.166 --> 00:02:30.383
or maybe this really is
the maximum distance.

00:02:30.383 --> 00:02:31.927
And if you're taking this
on a standardized test,

00:02:31.927 --> 00:02:34.377
this could become a very
involved thought process.

00:02:34.377 --> 00:02:35.736
And you start saying,
"well, how do I prove

00:02:35.736 --> 00:02:38.440
"that this is absolutely
the furthest distance?"

00:02:38.440 --> 00:02:41.005
But, many times, the
test itself is designed

00:02:41.005 --> 00:02:43.164
so that you don't have to prove it.

00:02:43.164 --> 00:02:44.789
Because notice, we could definitely find

00:02:44.789 --> 00:02:46.449
a distance of 22.

00:02:46.449 --> 00:02:48.574
This we can definitely
find between two points

00:02:48.574 --> 00:02:50.245
on these two surfaces.

00:02:50.245 --> 00:02:52.103
And notice, that is the largest of all

00:02:52.103 --> 00:02:53.705
of your possible choices.

00:02:53.705 --> 00:02:55.864
So, we would go with the largest

00:02:55.864 --> 00:02:58.047
of all of their possible choices.

00:02:58.047 --> 00:02:59.242
This would be a little bit more confusing.

00:02:59.242 --> 00:03:00.647
You would have to be a little bit more

00:03:00.647 --> 00:03:02.169
rigorous with yourself and say,

00:03:02.169 --> 00:03:03.732
"Well, do I really know that."

00:03:03.732 --> 00:03:06.150
If they had a choice that said 23

00:03:06.150 --> 00:03:08.367
or a choice that said 22 times

00:03:08.367 --> 00:03:11.223
the square root of two,
or something like that.

00:03:11.223 --> 00:03:13.046
But, the fact that you can definitely

00:03:13.046 --> 00:03:17.120
can get these points to be
22 away from each other,

00:03:17.120 --> 00:03:19.082
and that is the largest
of all of the choices.

00:03:19.082 --> 00:03:20.684
That's a pretty good clue that that is

00:03:20.684 --> 00:03:23.900
the maximum possible length of PQ.

00:03:23.900 --> 00:03:26.466
And in a future video, I
can do a more in depth proof

00:03:26.466 --> 00:03:29.888
that this is actually
going to be the case.


16 Compound inequality with absolute value

WEBVTT
Kind: captions
Language: en

00:00:00.566 --> 00:00:01.819
- [Voiceover] If six is less than

00:00:01.819 --> 00:00:04.106
the absolute value of x minus three,

00:00:04.106 --> 00:00:05.742
which is less than seven,

00:00:05.742 --> 00:00:07.868
and x is equal to zero,

00:00:07.868 --> 00:00:12.868
what is one possible value
of the absolute value of x?

00:00:13.660 --> 00:00:16.202
So let's just rewrite this
first compound inequality,

00:00:16.202 --> 00:00:17.527
then we'll think about this constrained

00:00:17.527 --> 00:00:19.256
of x being less than zero.

00:00:19.256 --> 00:00:21.578
So they're telling us that six is less

00:00:21.578 --> 00:00:25.003
than the absolute value of x minus three,

00:00:25.003 --> 00:00:27.661
which is less than seven.

00:00:27.661 --> 00:00:29.181
So let's just think about what it means

00:00:29.181 --> 00:00:30.796
to take the absolute value.

00:00:30.796 --> 00:00:32.757
An absolute value of an expression,

00:00:32.757 --> 00:00:34.383
or of a number could be viewed as,

00:00:34.383 --> 00:00:37.227
well, how far is that
number away from zero?

00:00:37.227 --> 00:00:39.305
For example, we've seen this before,

00:00:39.305 --> 00:00:43.437
negative five is five away from zero.

00:00:43.437 --> 00:00:46.526
Five is five away from zero.

00:00:46.526 --> 00:00:48.024
So if you think about it that way,

00:00:48.024 --> 00:00:49.045
now let's draw a number line,

00:00:49.045 --> 00:00:51.784
and visualize where
would x minus three sit?

00:00:51.784 --> 00:00:53.108
Where are all the possible places

00:00:53.108 --> 00:00:54.524
where x minus three could sit?

00:00:54.524 --> 00:00:55.966
I'm going to try and draw a pretty big

00:00:55.966 --> 00:01:00.117
number line so we can
visualize it reasonably well.

00:01:00.117 --> 00:01:03.890
So let's say that this
right over here is zero,

00:01:03.890 --> 00:01:07.071
this is positive six,

00:01:07.071 --> 00:01:11.247
this is positive seven.

00:01:11.247 --> 00:01:14.924
This is negative six,

00:01:14.924 --> 00:01:17.444
and this is negative seven.

00:01:17.444 --> 00:01:20.090
So if x minus three sits in this interval

00:01:20.090 --> 00:01:23.504
between six and seven,
and not equaling those,

00:01:23.504 --> 00:01:25.489
because these are not less than or equal,

00:01:25.489 --> 00:01:27.254
these are just less than signs.

00:01:27.254 --> 00:01:30.376
So if x minus three is in this interval

00:01:30.376 --> 00:01:34.195
between six and seven, so if x minus three

00:01:34.195 --> 00:01:36.970
is there, then the absolute value

00:01:36.970 --> 00:01:39.095
of x minus three is going to be between

00:01:39.095 --> 00:01:42.170
six and seven away from zero.

00:01:42.170 --> 00:01:45.073
But the other place where
x minus three could be,

00:01:45.073 --> 00:01:47.233
is in this interval,

00:01:47.233 --> 00:01:49.973
between negative six and negative seven.

00:01:49.973 --> 00:01:53.199
X minus three could also be there,

00:01:53.199 --> 00:01:55.592
because anything here,
if you take any value

00:01:55.592 --> 00:01:57.077
in that interval right over there,

00:01:57.077 --> 00:02:00.514
its absolute value,
its distance from zero,

00:02:00.514 --> 00:02:03.833
it's between six and seven away from zero.

00:02:03.833 --> 00:02:07.990
Notice something over here would be,

00:02:07.990 --> 00:02:10.195
so for example, negative 6.2,

00:02:10.195 --> 00:02:14.259
that is between six and
seven away from zero.

00:02:14.259 --> 00:02:16.069
So this is the absolute value,

00:02:16.069 --> 00:02:19.378
this distance to the left,

00:02:19.378 --> 00:02:21.271
if we're thinking about this area,

00:02:21.271 --> 00:02:24.150
and it's the distance
to the right of zero,

00:02:24.150 --> 00:02:26.495
if we're thinking about
things in this interval.

00:02:26.495 --> 00:02:28.119
So what are the two situations?

00:02:28.119 --> 00:02:30.696
Well, this blue situation is just

00:02:30.696 --> 00:02:34.447
six is less than x minus three,

00:02:34.447 --> 00:02:36.141
which is less than seven.

00:02:36.141 --> 00:02:38.499
We're essentially assuming
that x minus three

00:02:38.499 --> 00:02:40.820
is positive, but then there's a case

00:02:40.820 --> 00:02:42.979
where x minus three could be negative.

00:02:42.979 --> 00:02:47.956
And in that case, we have negative seven,

00:02:47.956 --> 00:02:52.956
is going to be less than x minus three,

00:02:53.473 --> 00:02:56.539
which is going to be
less than negative six.

00:02:56.539 --> 00:02:58.327
So these are the two cases

00:02:58.327 --> 00:03:01.160
and to figure out what x would
be in each of these cases,

00:03:01.160 --> 00:03:03.388
we could add three to all three parts

00:03:03.388 --> 00:03:05.107
of this inequality.

00:03:05.107 --> 00:03:08.021
So, if you add three to six,

00:03:08.021 --> 00:03:09.960
you get nine is less than...

00:03:09.960 --> 00:03:11.642
Add three here, that's the whole point,

00:03:11.642 --> 00:03:13.768
the three and the
negative three cancel out,

00:03:13.768 --> 00:03:16.379
is less than x, and then,

00:03:16.379 --> 00:03:20.130
less than seven plus three is 10.

00:03:20.130 --> 00:03:21.848
So that's one possibility.

00:03:21.848 --> 00:03:23.727
The other possibility over here,

00:03:23.727 --> 00:03:25.481
if you add three to
all the different parts

00:03:25.481 --> 00:03:27.699
of this compound inequality,

00:03:27.699 --> 00:03:30.565
negative seven plus
three is negative four,

00:03:30.565 --> 00:03:32.841
is less than x minus three, plus three,

00:03:32.841 --> 00:03:36.139
is x, which is less than negative six

00:03:36.139 --> 00:03:38.785
plus three, is negative three.

00:03:38.785 --> 00:03:41.502
Now, so, if this was our only constraint,

00:03:41.502 --> 00:03:44.091
this is what we would know about x,

00:03:44.091 --> 00:03:46.539
that either this is true, or this is true,

00:03:46.539 --> 00:03:48.398
but then they tell us another constraint.

00:03:48.398 --> 00:03:53.243
They tell us that x is less than...

00:03:53.243 --> 00:03:56.074
x is less than zero.

00:03:56.074 --> 00:03:58.956
So x can't be greater than nine,

00:03:58.956 --> 00:04:00.917
so we can rule out this case,

00:04:00.917 --> 00:04:02.995
which pretty much rules out that case,

00:04:02.995 --> 00:04:04.586
and makes us focus on this case,

00:04:04.586 --> 00:04:07.324
x has to be less than zero.

00:04:07.324 --> 00:04:10.541
So what's one possible value
of the absolute value of x?

00:04:10.541 --> 00:04:12.514
Well, you can pick any x in this interval

00:04:12.514 --> 00:04:14.894
between negative three and negative four.

00:04:14.894 --> 00:04:17.983
You could say, well, x could be equal

00:04:17.983 --> 00:04:20.919
to negative 3.1, you could say x

00:04:20.919 --> 00:04:23.542
is equal to negative 3.01,

00:04:23.542 --> 00:04:27.560
you could say x is equal to negative 3.56,

00:04:27.560 --> 00:04:32.482
you could say x is equal to negative 3.99.

00:04:32.482 --> 00:04:35.523
All of these x's are in this interval,

00:04:35.523 --> 00:04:37.301
so one possible value?

00:04:37.301 --> 00:04:41.212
Well, we could just say x
is equal to negative 3.5,

00:04:41.212 --> 00:04:43.081
you take the absolute value of x,

00:04:43.081 --> 00:04:45.856
so then, the absolute value of x

00:04:45.856 --> 00:04:49.757
would be equal to positive 3.5,

00:04:49.757 --> 00:04:52.339
and we are done.


5 Values on number line

WEBVTT
Kind: captions
Language: en

00:00:00.428 --> 00:00:01.949
- [Voiceover] On the number line above,

00:00:01.949 --> 00:00:03.852
the tick marks are equally spaced

00:00:03.852 --> 00:00:06.148
and their coordinates are shown.

00:00:06.349 --> 00:00:08.949
Of these coordinates,
which has the smallest

00:00:08.949 --> 00:00:11.130
positive value?

00:00:11.155 --> 00:00:11.898
So this is interesting.

00:00:11.898 --> 00:00:13.050
These are equally spaced.

00:00:13.050 --> 00:00:15.172
We start at negative eight and we have

00:00:15.172 --> 00:00:18.845
one, two, three, four,

00:00:18.845 --> 00:00:21.626
five, six equal spaces

00:00:21.626 --> 00:00:23.610
to get all the way to 10.

00:00:23.809 --> 00:00:26.849
So what must be the space
between these tick marks here?

00:00:26.965 --> 00:00:29.277
Well, to go from negative eight to 10,

00:00:29.277 --> 00:00:31.880
to go all the way from
negative eight to 10,

00:00:31.880 --> 00:00:34.152
you have increased by 18

00:00:34.152 --> 00:00:37.345
or the difference between
negative eight and 10 is 18.

00:00:37.345 --> 00:00:38.936
You have to go eight to get to zero

00:00:38.936 --> 00:00:41.873
and then 10 more to get to 10.

00:00:41.873 --> 00:00:43.125
So this distance is 18.

00:00:43.184 --> 00:00:45.390
We've split it into six sections,

00:00:45.390 --> 00:00:47.155
six equal sections.

00:00:47.155 --> 00:00:49.105
So 18 divided by six,

00:00:49.105 --> 00:00:52.493
each of these sections must
have a length of three.

00:00:52.866 --> 00:00:55.873
So they each have a length of three.

00:00:55.873 --> 00:00:57.173
So now we can figure out the values

00:00:57.173 --> 00:01:00.167
of A, B, C, D and E.

00:01:00.411 --> 00:01:02.652
A gets us to negative five.

00:01:02.652 --> 00:01:04.383
I'm just adding three each time.

00:01:04.383 --> 00:01:08.004
B is at negative two
and then you add three,

00:01:08.004 --> 00:01:10.187
you get to C, which is at one

00:01:10.187 --> 00:01:11.592
and we've already answered our question.

00:01:11.592 --> 00:01:13.333
This is the smallest positive value.

00:01:13.333 --> 00:01:15.783
We went from the most
negative, we keep increasing

00:01:15.783 --> 00:01:17.826
and this is the first of the choices.

00:01:17.826 --> 00:01:20.145
This is the first positive value we hit.

00:01:20.206 --> 00:01:24.925
So C has the smallest
positive, positive value.

00:01:24.925 --> 00:01:26.788
Then we could keep
going if you're curious.

00:01:26.788 --> 00:01:28.564
D, you're going to add another three.

00:01:28.564 --> 00:01:29.731
D is four.

00:01:29.731 --> 00:01:31.632
E is seven and then you add three more

00:01:31.632 --> 00:01:33.485
and we get to 10.

00:01:33.568 --> 00:01:36.677
We wanted the smallest positive value

00:01:36.677 --> 00:01:37.991
out of the choices,

00:01:37.991 --> 00:01:40.066
and that's going to be C.


8 Total videos rented

WEBVTT
Kind: captions
Language: en

00:00:00.351 --> 00:00:02.510
- [Voiceover] The first
table above shows the number

00:00:02.510 --> 00:00:05.971
of premium members at two
premium video rental stores.

00:00:05.971 --> 00:00:07.229
A number of premium members

00:00:07.229 --> 00:00:10.030
and then this is store A and store B.

00:00:10.030 --> 00:00:12.870
During the years two
thousand to two thousand two

00:00:12.870 --> 00:00:14.990
the second table shows the average,

00:00:14.990 --> 00:00:17.710
arithmetic mean, number of video rentals

00:00:17.710 --> 00:00:22.331
per premium member at store
B during each of those years.

00:00:22.331 --> 00:00:25.050
Based on this information,
which of the following

00:00:25.050 --> 00:00:29.051
best approximates the total
number of video rentals

00:00:29.051 --> 00:00:31.131
by, let me underline this,

00:00:31.131 --> 00:00:34.691
the total number of video rentals

00:00:34.691 --> 00:00:37.531
by premium members at store B

00:00:37.531 --> 00:00:40.891
during the years two
thousand to two thousand two?

00:00:40.891 --> 00:00:44.849
So, we wanna concern
ourselves with store B

00:00:44.849 --> 00:00:48.930
between two thousand and
two thousand and two.

00:00:48.930 --> 00:00:50.769
So, let's think about it a little bit.

00:00:50.769 --> 00:00:54.689
In the year two thousand
how many total rentals

00:00:54.689 --> 00:00:56.129
would there have been?

00:00:56.129 --> 00:00:58.088
Well, in the year two thousand you

00:00:58.088 --> 00:01:00.089
have five hundred premium members,

00:01:00.089 --> 00:01:02.409
you have five hundred premium members

00:01:02.409 --> 00:01:06.070
and on average they each rented twelve,

00:01:06.070 --> 00:01:08.671
twelve rentals per year,

00:01:08.671 --> 00:01:12.470
so, the total number of
rentals per premium member

00:01:12.470 --> 00:01:16.769
is going to be five hundred
times five hundred times twelve

00:01:16.769 --> 00:01:17.191
and you might be saying,

00:01:17.191 --> 00:01:20.900
"Hey, Sal, look, not all of
them rented exactly twelve."

00:01:20.900 --> 00:01:24.061
But you have to realize this
average is actually created

00:01:24.061 --> 00:01:27.861
by taking the sum of all
the five hundred's rentals

00:01:27.861 --> 00:01:30.022
and then dividing by five hundred.

00:01:30.022 --> 00:01:32.901
So, if you multiplied
by five hundred again

00:01:32.901 --> 00:01:35.521
then you're going to get the
total number that were rented.

00:01:35.521 --> 00:01:38.127
Let me just, let me just, let me just,

00:01:38.127 --> 00:01:40.733
let me just clarify that for you.

00:01:40.733 --> 00:01:43.340
So, if you took, if you knew
the total that were rented,

00:01:43.340 --> 00:01:46.801
total rented in two thousand,

00:01:46.801 --> 00:01:48.962
well, if you divided that by five hundred,

00:01:48.962 --> 00:01:51.262
if you divided that by five hundred,

00:01:51.262 --> 00:01:53.143
then you're going to get the average,

00:01:53.143 --> 00:01:54.544
you're going to get the average

00:01:54.544 --> 00:01:57.722
that were rented per or
the average videos rented

00:01:57.722 --> 00:01:59.263
per premium member.

00:01:59.263 --> 00:02:02.303
So, then if you multiply both
sides of this by five hundred,

00:02:02.303 --> 00:02:05.503
by five hundred, you're going
to get that the total rented

00:02:05.503 --> 00:02:07.799
is twelve times five hundred.

00:02:07.799 --> 00:02:09.841
And you could view it as on average

00:02:09.841 --> 00:02:12.424
twelve videos were rented per member

00:02:12.424 --> 00:02:14.103
times five hundred members

00:02:14.103 --> 00:02:16.783
and that's exactly what
we did right over there.

00:02:16.783 --> 00:02:18.642
Let me clear this up.

00:02:18.642 --> 00:02:21.041
And so let me clear that up.

00:02:21.041 --> 00:02:24.318
So, now we can think about
the year two thousand and one.

00:02:24.318 --> 00:02:27.583
So, in two thousand, two thousand and one,

00:02:27.583 --> 00:02:29.463
and it's the same logic.

00:02:29.463 --> 00:02:31.303
You have a thousand premium members

00:02:31.303 --> 00:02:35.223
and they each rented on
average fifteen videos each.

00:02:35.223 --> 00:02:38.821
So, it's going to be one
thousand times fifteen

00:02:38.821 --> 00:02:41.001
and then finally in two thousand and two,

00:02:41.001 --> 00:02:44.652
two thousand and two you have one thousand

00:02:44.652 --> 00:02:46.591
two hundred and fifty premium members

00:02:46.591 --> 00:02:49.991
and they each rented twenty videos each

00:02:49.991 --> 00:02:52.230
or on average, twenty each.

00:02:52.230 --> 00:02:54.230
So, if you sum these up
what're you going to get?

00:02:54.230 --> 00:02:56.251
Let's see, five hundred times twelve.

00:02:56.251 --> 00:02:57.492
Five times twelve is sixty.

00:02:57.492 --> 00:02:59.654
Sixty, and then you have two more zeros.

00:02:59.654 --> 00:03:01.286
So, that's six thousand.

00:03:01.286 --> 00:03:05.599
One thousand times fifteen
is fifteen thousand

00:03:05.599 --> 00:03:07.300
and then, let's see,
one thousand two hundred

00:03:07.300 --> 00:03:11.264
and fifty times two would
be twenty five hundred

00:03:11.264 --> 00:03:13.464
but then you have another zero
so it's twenty five thousand,

00:03:13.464 --> 00:03:17.123
twenty five thousand, you
add these all together

00:03:17.123 --> 00:03:17.825
and what do you get?

00:03:17.825 --> 00:03:20.098
Six thousand plus fifteen thousand is

00:03:20.098 --> 00:03:23.465
twenty one thousand plus
twenty five thousand

00:03:23.465 --> 00:03:27.423
is equal to forty six, forty six thousand

00:03:27.423 --> 00:03:30.060
which is choice B.

00:03:30.060 --> 00:03:32.194
Choice B right over there.


18 Area of rectangle

WEBVTT
Kind: captions
Language: en

00:00:00.694 --> 00:00:04.164
- [Voiceover] In the figure
above, PQRS is a rectangle.

00:00:04.164 --> 00:00:06.056
So that, let us note that these are

00:00:06.056 --> 00:00:08.530
right angles right over there.

00:00:08.530 --> 00:00:10.987
The area of triangle RST is seven.

00:00:10.987 --> 00:00:13.143
So RST is seven.

00:00:13.143 --> 00:00:16.405
And PT is equal to 2/5 PS.

00:00:16.405 --> 00:00:19.981
What is the area of triangle PQRS?

00:00:19.981 --> 00:00:22.175
So let's think about what
information they've given us

00:00:22.175 --> 00:00:24.044
and what we need to figure out.

00:00:24.044 --> 00:00:29.044
So they tell us the area
of triangle RST is seven.

00:00:29.500 --> 00:00:31.764
So what is the area of triangle RST,

00:00:31.764 --> 00:00:34.225
given these lengths right over here?

00:00:34.225 --> 00:00:37.197
Well, triangle RST, the
area of any triangle is

00:00:37.197 --> 00:00:39.960
1/2 times the base times the height.

00:00:39.960 --> 00:00:41.527
So what is the base of RST?

00:00:41.527 --> 00:00:45.033
What we could view it as
the length of segment ST.

00:00:45.033 --> 00:00:48.852
So the area is going to be equal to 1/2

00:00:48.852 --> 00:00:53.852
times the length of segment ST,

00:00:54.329 --> 00:00:56.543
for that (mumbles) times the height

00:00:56.543 --> 00:01:01.543
which is the length of segment RS.

00:01:01.743 --> 00:01:06.166
So times RS and that's equal to seven.

00:01:06.166 --> 00:01:08.423
And that's equal to seven.

00:01:08.423 --> 00:01:10.431
And if we wanted to
simplify this a little bit,

00:01:10.431 --> 00:01:13.167
we could multiply both sides by two.

00:01:13.167 --> 00:01:16.974
And so, we could get the
length of segment ST times

00:01:16.974 --> 00:01:21.974
the length of segment RS is equal to 14.

00:01:22.268 --> 00:01:23.886
Just multiply both sides by two.

00:01:23.886 --> 00:01:25.217
Two times 1/2 is one.

00:01:25.217 --> 00:01:27.473
Two times seven is 14.

00:01:27.473 --> 00:01:30.155
So this first piece of information is

00:01:30.155 --> 00:01:33.537
that the product of ST and RS,

00:01:33.537 --> 00:01:36.877
ST and RS is 14.

00:01:36.877 --> 00:01:41.613
Then they tell us that
PT, I'll do PT in orange,

00:01:41.613 --> 00:01:45.432
then they tell us that PT is

00:01:45.432 --> 00:01:49.811
equal to 2/5

00:01:49.811 --> 00:01:51.690
times PS.

00:01:51.690 --> 00:01:54.650
Now let's just say,
I'm gonna use pink for,

00:01:54.650 --> 00:01:57.171
I'll just use this dark blue for PS.

00:01:57.171 --> 00:01:58.998
Times PS.

00:01:58.998 --> 00:02:01.544
So that's that length right
over there, that's PS.

00:02:01.544 --> 00:02:04.136
2/5 times PS.

00:02:04.136 --> 00:02:06.771
So that's the second piece of information.

00:02:06.771 --> 00:02:09.511
Right over here, I just rewrote it.

00:02:09.511 --> 00:02:11.325
And now, what are we gonna figure out?

00:02:11.325 --> 00:02:14.813
We need the area of PQRS
of this entire rectangle.

00:02:14.813 --> 00:02:16.520
Well, the area of rectangle is going to be

00:02:16.520 --> 00:02:18.082
its base times its height.

00:02:18.082 --> 00:02:20.947
So the area, what we need to figure out is

00:02:20.947 --> 00:02:24.128
going to be PS, the length of segment PS.

00:02:24.128 --> 00:02:27.506
So I can write, so we
need to figure out PS

00:02:27.506 --> 00:02:32.374
times the height which is RS.

00:02:32.374 --> 00:02:33.682
Times RS.

00:02:33.682 --> 00:02:37.791
This is what we need to figure out.

00:02:37.791 --> 00:02:41.112
So going back to what we
know, so here we know PT

00:02:41.112 --> 00:02:44.792
in terms of PS or PS in terms of PT,

00:02:44.792 --> 00:02:49.017
here we have an expression
with ST, RS, and 14.

00:02:49.017 --> 00:02:52.152
So somehow, if I could turn
this ST part of this expression,

00:02:52.152 --> 00:02:56.215
if I could turn this into
something that deals with PS,

00:02:56.215 --> 00:02:58.316
then I'm gonna be in pretty good shape.

00:02:58.316 --> 00:03:00.418
Now let's just look at this.

00:03:00.418 --> 00:03:03.610
How does ST relate to PS?

00:03:03.610 --> 00:03:05.154
So let me write this over here.

00:03:05.154 --> 00:03:07.835
So S, the length of segment ST

00:03:07.835 --> 00:03:10.963
is equal to the length of segment PS.

00:03:10.963 --> 00:03:13.488
It's equal to the length of segment PS.

00:03:13.488 --> 00:03:18.011
Minus the length of segment PT.

00:03:18.011 --> 00:03:21.012
Minus the length of segment PT.

00:03:21.012 --> 00:03:23.003
So I could substitute this in for ST

00:03:23.003 --> 00:03:26.741
but now it has something as
in terms of both PS and PT.

00:03:26.741 --> 00:03:28.638
But lucky for us, we
can also substitute out

00:03:28.638 --> 00:03:30.592
the PT for something in terms of PS.

00:03:30.592 --> 00:03:32.624
That's what this does right over here.

00:03:32.624 --> 00:03:37.602
So this, we know that PT, the
second piece of information

00:03:37.602 --> 00:03:40.523
they've given us, PT is equal to 2/5 PS.

00:03:40.523 --> 00:03:43.728
So we can rewrite this
whole thing and saying ST,

00:03:43.728 --> 00:03:45.964
the length of segment ST is equal

00:03:45.964 --> 00:03:50.439
to the length of segment PS minus,

00:03:50.439 --> 00:03:55.439
minus 2/5 times

00:03:55.695 --> 00:03:58.341
the length of segment PS.

00:03:58.341 --> 00:04:01.165
Minus 2/5 times the length of segment PS.

00:04:01.165 --> 00:04:03.220
And once again, how did I do that?

00:04:03.220 --> 00:04:06.397
Well, we know that that
is equivalent to that

00:04:06.397 --> 00:04:08.236
from this piece of information,

00:04:08.236 --> 00:04:10.383
from this piece of information here.

00:04:10.383 --> 00:04:11.694
Now what's powerful about this is

00:04:11.694 --> 00:04:14.662
now I have ST purely in terms of PS.

00:04:14.662 --> 00:04:18.026
So if I replace ST with
what I have in terms of PS,

00:04:18.026 --> 00:04:19.164
it looks like I'm gonna have some type

00:04:19.164 --> 00:04:21.901
of product of PS and RS or something

00:04:21.901 --> 00:04:25.335
that I can solve for the
product of PS and RS.

00:04:25.335 --> 00:04:28.222
Before I even do that, let's
simplify this a little bit.

00:04:28.222 --> 00:04:31.105
If I have one PS minus 2/5 PS

00:04:31.105 --> 00:04:35.145
or we could even say 5/5 PS minus 2/5 PS,

00:04:35.145 --> 00:04:40.143
we get ST is equal to 3/5,

00:04:40.143 --> 00:04:45.143
3/5 times PS.

00:04:45.222 --> 00:04:47.284
And now we are ready to substitute.

00:04:47.284 --> 00:04:49.099
So instead of having an ST here,

00:04:49.099 --> 00:04:54.099
we could write 3/5 PS

00:04:54.126 --> 00:04:57.666
3/5 PS

00:04:57.666 --> 00:05:00.128
times the length of segment RS.

00:05:00.128 --> 00:05:01.075
These are all the, you know the...

00:05:01.075 --> 00:05:02.657
3/5 times the length of segment PS

00:05:02.657 --> 00:05:07.405
times the length of
segment RS is equal to 14.

00:05:07.405 --> 00:05:09.193
And now we could just
solve for this product.

00:05:09.193 --> 00:05:10.586
What is PS times RS?

00:05:10.586 --> 00:05:12.138
Well, let's just multiply both sides

00:05:12.138 --> 00:05:14.753
times the reciprocal of 3/5.

00:05:14.753 --> 00:05:17.923
Times 5 over three.

00:05:17.923 --> 00:05:21.994
And we get, I think we
deserve a drum roll now,

00:05:21.994 --> 00:05:25.233
we get the product of PS

00:05:25.233 --> 00:05:29.880
and RS being equal to,

00:05:29.880 --> 00:05:32.578
let's see, what's 14 times five?

00:05:32.578 --> 00:05:34.570
It's gonna be 50 plus 20.

00:05:34.570 --> 00:05:35.998
It's going to be 70.

00:05:35.998 --> 00:05:40.781
So this is going to be
equal to 70 over three.

00:05:40.781 --> 00:05:43.516
And we are done.


14 Small arc length

WEBVTT
Kind: captions
Language: en

00:00:00.834 --> 00:00:02.767
- [voiceover] Point O is the center

00:00:02.767 --> 00:00:05.167
of both circles in the figure above.

00:00:05.167 --> 00:00:08.566
If the circumference of
the larger circle is 36,

00:00:08.566 --> 00:00:12.640
so the circumference of the
larger circle there is 36,

00:00:12.640 --> 00:00:16.900
and the radius of the small
circle is half the radius

00:00:16.900 --> 00:00:19.834
of the larger circle, what is the length

00:00:19.834 --> 00:00:22.233
of the darkened arc?

00:00:22.233 --> 00:00:23.900
If you have half the radius,

00:00:23.900 --> 00:00:25.703
you're going to have
half the circumference.

00:00:25.703 --> 00:00:28.566
We could even solve for
what that radius is.

00:00:28.566 --> 00:00:31.033
You want to know what the
radius of the larger circle is?

00:00:31.033 --> 00:00:36.033
The larger circle, we could
say, two pi times its radius

00:00:37.034 --> 00:00:41.048
is equal to 36, and so its radius,

00:00:41.048 --> 00:00:43.700
if you divide both sides by two pi,

00:00:43.700 --> 00:00:46.700
it's radius is going to be equal to,

00:00:46.700 --> 00:00:49.966
36 divided by two is 18 over pi.

00:00:49.966 --> 00:00:51.766
That's the larger circle.

00:00:51.766 --> 00:00:53.100
The smaller circle has a radius

00:00:53.100 --> 00:00:55.633
half of the radius of the larger circle.

00:00:55.633 --> 00:00:59.500
The smaller circle's radius is, so radius,

00:00:59.500 --> 00:01:01.633
and I'll write a little
sub S there for the radius

00:01:01.633 --> 00:01:04.033
of the smaller circle is
going to be half this.

00:01:04.033 --> 00:01:06.233
It's going to be nine over pi.

00:01:06.233 --> 00:01:08.700
The circumference of the smaller circle,

00:01:08.700 --> 00:01:11.433
the circumference of the
smaller circle is going to be

00:01:11.433 --> 00:01:16.100
its radius, nine over pi times two pi.

00:01:16.100 --> 00:01:18.366
Pi divided by pi.

00:01:18.366 --> 00:01:23.100
You're going to have nine
times two is equal to 18.

00:01:23.100 --> 00:01:26.233
The circumference here is 18.

00:01:26.233 --> 00:01:28.033
It might have already jumped out at you

00:01:28.033 --> 00:01:31.234
because circumference is
directly proportional to,

00:01:31.234 --> 00:01:33.566
because circumference for any circle

00:01:33.566 --> 00:01:36.233
is directly proportional to a radius,

00:01:36.233 --> 00:01:39.034
if you have half the
radius, you're going to have

00:01:39.034 --> 00:01:40.633
half the circumference and you would not

00:01:40.633 --> 00:01:41.833
have to go through all of this.

00:01:41.833 --> 00:01:43.968
You could've said, "Hey,
okay, if I have a radius here

00:01:43.968 --> 00:01:47.033
"of 36 and if I have a
circumference here of 36

00:01:47.033 --> 00:01:48.966
"and if I have a circle
with half the radius,

00:01:48.966 --> 00:01:51.100
"its circumference is going to half that."

00:01:51.100 --> 00:01:53.033
It's going to be 18.

00:01:53.033 --> 00:01:54.166
That's not what they're asking us.

00:01:54.166 --> 00:01:56.166
They're not asking us
just the circumference

00:01:56.166 --> 00:01:57.900
of the smaller circle.

00:01:57.900 --> 00:02:00.433
They're asking us what is the arc length.

00:02:00.433 --> 00:02:03.300
What is the arc length of
this part right over here?

00:02:03.300 --> 00:02:04.966
This 80 degrees.

00:02:04.966 --> 00:02:07.833
80 degrees, remember,
to go all the way around

00:02:07.833 --> 00:02:09.766
the circle is 360 degrees.

00:02:09.766 --> 00:02:14.766
This is 80/360 of the entire
circle right over here,

00:02:15.833 --> 00:02:17.567
or the entire circumference.

00:02:17.567 --> 00:02:19.366
If you want that arc
length, it's going to be

00:02:19.366 --> 00:02:24.366
the circumference times
80/360, times 80 over 360.

00:02:27.767 --> 00:02:29.567
That's the same thing as, let's see,

00:02:29.567 --> 00:02:31.034
then divide both of them by 10.

00:02:31.034 --> 00:02:32.966
That's the same thing as 8/36.

00:02:32.966 --> 00:02:33.700
Let's see.

00:02:33.700 --> 00:02:35.967
You can divide eight
divided by four is two,

00:02:35.967 --> 00:02:39.633
and then 36 divided by four is nine.

00:02:39.633 --> 00:02:41.633
Then you are left with, let's see,

00:02:41.633 --> 00:02:45.300
18 divided by nine is
two, so you're left with

00:02:45.300 --> 00:02:50.200
two times two is equal
to four, and you're done.


9 Similar triangles

WEBVTT
Kind: captions
Language: en

00:00:00.566 --> 00:00:04.166
- In triangle ABC above, AB,

00:00:04.166 --> 00:00:08.433
the length of this
segment, is equal to AC.

00:00:08.433 --> 00:00:10.166
E is the midpoint of AB.

00:00:10.166 --> 00:00:11.633
Well they really haven't
drawn this to scale.

00:00:11.633 --> 00:00:15.092
Let me see if I can draw a
little bit more to scale.

00:00:16.033 --> 00:00:19.104
So they're saying E is the midpoint of AB.

00:00:19.104 --> 00:00:22.767
So the midpoint looks
closer to around there.

00:00:22.767 --> 00:00:23.966
So it would look like this.

00:00:23.966 --> 00:00:28.766
This is A, this is E, this is B,

00:00:28.766 --> 00:00:31.633
that is D and that is C.

00:00:31.633 --> 00:00:32.833
So let me mark this off.

00:00:32.833 --> 00:00:35.833
E is the midpoint of AB

00:00:35.833 --> 00:00:39.572
so that means AE is equal to EB.

00:00:39.572 --> 00:00:42.633
And D is the mid point of AC.

00:00:42.633 --> 00:00:45.567
We know that AC is the same length as AB

00:00:45.567 --> 00:00:49.167
so AD is going to be
the same length as that

00:00:49.167 --> 00:00:50.769
which is also the same length as that

00:00:50.769 --> 00:00:52.767
which is the same length as that.

00:00:52.767 --> 00:00:55.433
If AE is equal to x--

00:00:55.433 --> 00:01:00.033
So AE, this distance right
over here is equal to x.

00:01:00.033 --> 00:01:02.635
And ED is equal to 4.

00:01:02.635 --> 00:01:06.966
So this is ED is equal to 4

00:01:06.966 --> 00:01:11.100
so this length of this
segment right over here is 4.

00:01:11.100 --> 00:01:13.611
What is the length of BC?

00:01:13.611 --> 00:01:16.367
What is the length of BC here?

00:01:16.367 --> 00:01:18.633
Now the key thing that
might jump out at you

00:01:18.633 --> 00:01:20.966
is that we're dealing with
two similar triangles.

00:01:20.966 --> 00:01:23.366
That triangle AED

00:01:23.366 --> 00:01:27.100
is similar to triangle ABC.

00:01:27.100 --> 00:01:28.766
And how do we know that they are similar?

00:01:28.766 --> 00:01:31.500
Well, their corresponding
sides have the same ratio.

00:01:31.500 --> 00:01:35.166
The ratio AC is twice AD,

00:01:35.166 --> 00:01:38.301
AB is twice AE

00:01:38.301 --> 00:01:42.766
and then they have a
common angle in between.

00:01:42.766 --> 00:01:45.967
So AED is similar to triangle ABC.

00:01:45.967 --> 00:01:47.500
Let me right this down.

00:01:47.500 --> 00:01:51.166
So triangle AED

00:01:51.166 --> 00:01:54.299
is similar

00:01:54.299 --> 00:01:58.972
to triangle ABC.

00:01:58.972 --> 00:02:01.500
And what that tells us
is the ratio between

00:02:01.500 --> 00:02:03.972
corresponding sides is
going to be the same.

00:02:03.972 --> 00:02:07.767
Now, BC is the corresponding
side on the larger triangle,

00:02:07.767 --> 00:02:12.433
on triangle ABC, to this side, side ED

00:02:12.433 --> 00:02:15.168
on the smaller triangle, on triangle AED.

00:02:15.168 --> 00:02:17.100
Now what's the ratio between

00:02:17.100 --> 00:02:20.033
corresponding sides of ABC and AED?

00:02:20.033 --> 00:02:23.700
Well, this length is x,

00:02:23.700 --> 00:02:25.233
we've already been told that,

00:02:25.233 --> 00:02:27.167
this length we know is two x.

00:02:27.167 --> 00:02:29.434
How do we know that?
Because E is the midpoint.

00:02:29.434 --> 00:02:32.100
So the ratio between
the corresponding sides

00:02:32.100 --> 00:02:34.233
of the larger triangle
and the smaller triangle,

00:02:34.233 --> 00:02:38.700
the ratio is two x to x
or the ratio is just two.

00:02:38.700 --> 00:02:41.966
Or another way of saying it
is the corresponding side

00:02:41.966 --> 00:02:43.633
of the larger triangle is twice

00:02:43.633 --> 00:02:47.300
the corresponding side
of the smaller triangle.

00:02:47.300 --> 00:02:51.300
So if this is four, this is
just going to be twice four.

00:02:51.300 --> 00:02:52.500
It's going to be eight.

00:02:52.500 --> 00:02:55.515
So the length of segment BC is eight.


1 Running distance

WEBVTT
Kind: captions
Language: en

00:00:00.531 --> 00:00:01.502
- [Voiceover] On Wednesday,

00:00:01.502 --> 00:00:06.502
Heather ran three miles in 30 minutes.

00:00:06.702 --> 00:00:11.702
If she ran for 45 minutes
at this rate on Thursday,

00:00:11.882 --> 00:00:14.242
how for did Heather run on Thursday?

00:00:14.242 --> 00:00:16.402
Well, let's just think
about what that rate is.

00:00:16.402 --> 00:00:20.446
So, the rate is going to
be equal to the distance,

00:00:20.446 --> 00:00:24.362
the distance she traveled per time,

00:00:24.362 --> 00:00:26.580
and this is going to
be equal to, let's see,

00:00:26.580 --> 00:00:28.822
she traveled three miles,

00:00:28.822 --> 00:00:33.822
she traveled three miles in
30 minutes, in 30 minutes.

00:00:37.018 --> 00:00:39.978
You could simplify, three
over 30 is the same thing,

00:00:39.978 --> 00:00:44.978
this is the same thing
as 1/10, 1/10 of a mile,

00:00:45.482 --> 00:00:50.482
of a mile per minute, per
minute, that's her rate.

00:00:50.598 --> 00:00:51.847
Now that's not what they are asking for,

00:00:51.847 --> 00:00:55.502
they're saying, if she went
at this rate for 45 minutes,

00:00:55.502 --> 00:00:57.482
how far would she run?

00:00:57.482 --> 00:01:02.482
Well, let's just multiply
them, so 45 minutes times,

00:01:03.482 --> 00:01:07.942
times, 1/10 of a mile per minute.

00:01:07.942 --> 00:01:10.182
I can just copy and paste
that right over there,

00:01:10.182 --> 00:01:15.182
so times that, copy and paste,
what are we going to get?

00:01:15.882 --> 00:01:18.442
Well it's going to be 45 times 1/10,

00:01:18.442 --> 00:01:21.522
or it's the same thing
as 45 divided by 10,

00:01:21.522 --> 00:01:25.514
that's going to be equal to
four point five, four point five

00:01:25.514 --> 00:01:28.161
and then the minutes cancel
out, minutes of the numerator,

00:01:28.161 --> 00:01:30.581
minutes of the denominator,
four point five,

00:01:30.581 --> 00:01:33.247
four point five miles.

00:01:33.247 --> 00:01:34.474
Another way to think about it,

00:01:34.474 --> 00:01:37.338
and this might even be an
easier way to think about it is,

00:01:37.338 --> 00:01:39.438
well she's going to get the
same rate, if you're going

00:01:39.438 --> 00:01:43.078
from 30 minutes on one day,
to 45 minutes on the next day,

00:01:43.078 --> 00:01:47.079
you're increasing, you're
increasing your minutes by 50%.

00:01:47.079 --> 00:01:50.353
Thirty, plus half of 30, 30 plus 15 is 45,

00:01:50.353 --> 00:01:53.925
so the distance you travel is
going to be increasing by 50%.

00:01:53.925 --> 00:01:58.925
So 50% increase on three,
would get you to, 50% of three

00:02:00.223 --> 00:02:01.918
is one point five, you add that to three,

00:02:01.918 --> 00:02:04.049
it gets you to four point five.

00:02:04.049 --> 00:02:06.783
So, either way you want to tackle it.


12 Missing angle

WEBVTT
Kind: captions
Language: en

00:00:00.307 --> 00:00:02.408
- [Voiceover] If x is equal to 20,

00:00:02.408 --> 00:00:04.797
so this right over here
is a 20 degree angle

00:00:04.797 --> 00:00:07.547
and this one also is a 20 degree angle,

00:00:07.547 --> 00:00:09.876
and y is equal to 30,

00:00:09.876 --> 00:00:12.351
so this right over here, we'll
do this a different color,

00:00:12.812 --> 00:00:15.009
this right over here is,

00:00:15.009 --> 00:00:16.378
I'm having trouble switching colors,

00:00:16.378 --> 00:00:18.369
that's a 30 degree angle.

00:00:18.369 --> 00:00:21.863
In the figure above, what is
the value of z right over here?

00:00:21.863 --> 00:00:23.618
This is a little bit strange at first

00:00:23.618 --> 00:00:27.042
because it feels like
we might be able to use

00:00:27.042 --> 00:00:32.042
triangles somehow, but this
whole shape is not a triangle.

00:00:32.225 --> 00:00:34.684
But, if we could construct a triangle or

00:00:34.684 --> 00:00:36.260
we know two of the angles of the triangle,

00:00:36.260 --> 00:00:38.318
then we know we can always
figure out the third

00:00:38.318 --> 00:00:39.948
because we know the sum
of the interior angles

00:00:39.948 --> 00:00:41.998
of a triangle add up to 180.

00:00:41.998 --> 00:00:44.529
So, can we construct a triangle where

00:00:44.529 --> 00:00:46.525
we know two of the angles?

00:00:46.525 --> 00:00:47.768
So, let's think about it.

00:00:47.768 --> 00:00:49.943
So, this is, we know this is 30 degrees.

00:00:49.943 --> 00:00:50.918
We know this is 20 degrees.

00:00:50.918 --> 00:00:52.637
Can we construct a
triangle where these are

00:00:52.637 --> 00:00:54.744
two of the angles of that triangle?

00:00:55.713 --> 00:00:56.968
Well, what if we,

00:00:57.398 --> 00:00:59.811
what if we extend this side

00:01:00.395 --> 00:01:01.807
to go all the way over here?

00:01:01.807 --> 00:01:04.390
Now we have, we know two of
the angles of a triangle.

00:01:04.390 --> 00:01:06.478
Now, we should be able to figure out

00:01:06.478 --> 00:01:08.431
this third angle right over here

00:01:08.431 --> 00:01:10.143
and what is a third angle going to be?

00:01:10.636 --> 00:01:13.617
Well, 30 degrees plus
this angle plus 20 degrees

00:01:13.617 --> 00:01:14.773
is going to be 180,

00:01:14.773 --> 00:01:16.111
or another way to think about it,

00:01:16.111 --> 00:01:18.598
this is going to be 180 degrees,

00:01:18.598 --> 00:01:21.292
minus 30, minus 20

00:01:21.292 --> 00:01:24.711
or this is going to be
180 degrees minus 50.

00:01:25.219 --> 00:01:29.744
So, this is going to be 130 degrees.

00:01:29.744 --> 00:01:32.222
Now, if we know that, we can figure out

00:01:32.222 --> 00:01:34.153
what this angle's going to be because

00:01:34.153 --> 00:01:35.820
these two angles are supplementary.

00:01:35.820 --> 00:01:38.196
They are going to add up to 180 degrees.

00:01:38.196 --> 00:01:39.923
When you go all the way
around, you've essentially,

00:01:39.923 --> 00:01:42.042
you've gone all the way
or you go halfway around

00:01:42.042 --> 00:01:43.648
the circle or you can kinda say

00:01:43.648 --> 00:01:46.241
the two rays of the angle form a line.

00:01:46.241 --> 00:01:49.167
And so, what plus 130 is 180.

00:01:49.167 --> 00:01:52.444
Well, 180 minus 130 is 50 degrees.

00:01:53.167 --> 00:01:54.774
So, this is 50 degrees, what do we know,

00:01:54.774 --> 00:01:59.252
what do we know about this angle

00:01:59.252 --> 00:02:00.992
right over here?

00:02:01.300 --> 00:02:05.949
Well, 50 plus 20 plus
this is going to be 180.

00:02:05.949 --> 00:02:07.634
So, another way of thinking about it,

00:02:07.634 --> 00:02:10.133
this is going to be 180 minus 50,

00:02:10.133 --> 00:02:14.583
minus 20 or 180 minus,
what is this, minus 70.

00:02:14.583 --> 00:02:17.630
So, this is going to be 110 degrees.

00:02:17.630 --> 00:02:19.256
Well, if that's 110 degrees,

00:02:19.256 --> 00:02:21.227
now we're really cutting
to the chase 'cause this,

00:02:21.227 --> 00:02:24.599
that angle, that angle is supplementary

00:02:24.599 --> 00:02:28.601
with the angle that has measure z degrees.

00:02:28.601 --> 00:02:33.601
So, we know z plus 110, z plus 110

00:02:33.979 --> 00:02:36.961
is going to be equal to 180

00:02:36.961 --> 00:02:39.697
or subtract 110 from both sides.

00:02:39.697 --> 00:02:44.144
You get z is equal to 70

00:02:44.144 --> 00:02:45.790
and we're done.


15 Equation of line

WEBVTT
Kind: captions
Language: en

00:00:00.604 --> 00:00:02.357
- [Voiceover] The graph
above shows a number

00:00:02.357 --> 00:00:06.850
of George's unsold candy
bars over a 10-day period.

00:00:06.850 --> 00:00:08.498
The points on the graph all lie

00:00:08.498 --> 00:00:10.167
on which of the following lines?

00:00:10.167 --> 00:00:12.891
Let's see, as the days go
by he has fewer and fewer

00:00:12.891 --> 00:00:16.568
unsold candy bars, all the
way until we get to day 10,

00:00:16.568 --> 00:00:18.187
and he sold all of his candy bars,

00:00:18.187 --> 00:00:20.507
'cause he has no unsold candy bars left.

00:00:20.507 --> 00:00:22.257
But anyway, they want us to figure out

00:00:22.257 --> 00:00:24.530
what could be the graph of the line,

00:00:24.530 --> 00:00:27.467
or the points on the graph all lie

00:00:27.467 --> 00:00:28.908
on which of the following lines?

00:00:28.908 --> 00:00:32.949
Let's just think about
this line a little bit.

00:00:32.949 --> 00:00:35.844
Let's just think about
this line a little bit.

00:00:35.844 --> 00:00:37.678
So when we're thinking about
any line, we could think about,

00:00:37.678 --> 00:00:42.182
"Well, what's the slope
of this line going to be?"

00:00:42.182 --> 00:00:45.215
So, let's see, our slope.

00:00:45.215 --> 00:00:48.853
Our slope is our change
in our vertical-axis

00:00:48.853 --> 00:00:51.872
for a given change in our horizontal-axis.

00:00:51.872 --> 00:00:53.787
And so let's say that we pick the change

00:00:53.787 --> 00:00:58.319
in our horizontal-axis
to be 10, 10 days go by.

00:00:58.319 --> 00:01:01.263
So our change in our
horizontal-axis is 10 days,

00:01:01.263 --> 00:01:03.226
and so what was the
change in our vertical?

00:01:03.226 --> 00:01:04.402
Well, the change in our vertical

00:01:04.402 --> 00:01:08.182
was 120 candy bars, or negative 120.

00:01:08.182 --> 00:01:11.630
So we went down, when number
of days increased by 10,

00:01:11.630 --> 00:01:15.612
positive 10, our candy
bars decreased by 120.

00:01:15.612 --> 00:01:18.549
Decreased by 120, and so
what's that going to be?

00:01:18.549 --> 00:01:22.714
That's going to be negative 12.

00:01:22.714 --> 00:01:25.858
So we want to align with
a slope of negative 12,

00:01:25.858 --> 00:01:28.770
and immediately when you look over here...

00:01:28.770 --> 00:01:30.890
we could already start to
think about the y-intercept,

00:01:30.890 --> 00:01:32.372
but immediately when you look over here,

00:01:32.372 --> 00:01:35.289
the slope is going to be
the coefficient on the x.

00:01:35.289 --> 00:01:37.925
So here this is a slope of
positive 10, that's not right.

00:01:37.925 --> 00:01:39.930
Slope of positive 10, that's not right.

00:01:39.930 --> 00:01:43.683
Slope of positive 12, not
negative 12, that's not right.

00:01:43.683 --> 00:01:46.724
Slope of negative 10, that's not right.

00:01:46.724 --> 00:01:48.749
And right over here, we
have our last choice,

00:01:48.749 --> 00:01:52.041
slope of negative 12, so our choice is E.

00:01:52.041 --> 00:01:54.665
But we could keep going even
if this was a free answer,

00:01:54.665 --> 00:01:57.625
we could still figure
this out, because we know

00:01:57.625 --> 00:02:00.168
that the slope of a line,
or the equation of a line,

00:02:00.168 --> 00:02:03.462
is y is equal to your slope...

00:02:03.462 --> 00:02:05.580
your slope times x,

00:02:05.580 --> 00:02:08.866
plus your y-intercept.

00:02:08.866 --> 00:02:11.518
Now we already figured out
that the slope right over here

00:02:11.518 --> 00:02:14.316
is negative 12, and
what's our y-intercept?

00:02:14.316 --> 00:02:16.825
Well, that's what y is equal
to when x is equal to zero.

00:02:16.825 --> 00:02:20.318
When x is equal to zero,
when no days have gone by,

00:02:20.318 --> 00:02:22.837
George has got 120 candy bars.

00:02:22.837 --> 00:02:26.041
So this right over here
is going to be 120.

00:02:26.041 --> 00:02:29.151
So the equation of our line is
going to be, y is equal to...

00:02:29.151 --> 00:02:31.353
even if we didn't have these choices,

00:02:31.353 --> 00:02:35.993
y is equal to negative 12 times x,

00:02:35.993 --> 00:02:38.298
plus 120.

00:02:38.298 --> 00:02:40.386
Or, you could obviously
swap these two around,

00:02:40.386 --> 00:02:42.878
and write it as 120 minus 12x,

00:02:42.878 --> 00:02:46.106
which is exactly what
they did right over here.


6 Possible median value

WEBVTT
Kind: captions
Language: en

00:00:00.659 --> 00:00:03.131
- [Voiceover] If x is the
median of the seven numbers

00:00:03.131 --> 00:00:05.231
listed above, which of the following

00:00:05.231 --> 00:00:07.809
could be the value of x?

00:00:07.809 --> 00:00:09.251
So the median means that if we were to put

00:00:09.251 --> 00:00:11.689
these in order, x would
be the middle number.

00:00:11.689 --> 00:00:13.178
We have seven numbers,
so it would actually be

00:00:13.178 --> 00:00:14.524
the exact middle number.

00:00:14.524 --> 00:00:15.983
If we had an even number, then it would be

00:00:15.983 --> 00:00:17.759
the middle two averaged.

00:00:17.759 --> 00:00:20.453
So let's think about this.

00:00:20.453 --> 00:00:21.800
Let's order this and let's put x

00:00:21.800 --> 00:00:23.262
in the middle of that order.

00:00:23.262 --> 00:00:26.594
So the smallest number here is three,

00:00:26.594 --> 00:00:31.594
then we go to four, then the next one,

00:00:32.851 --> 00:00:34.514
let's see, we go to 10.

00:00:35.050 --> 00:00:36.291
So those would be the three numbers

00:00:36.291 --> 00:00:37.885
that are less than x.

00:00:37.885 --> 00:00:39.323
Then we go to x, and now we go to

00:00:39.323 --> 00:00:41.766
the three numbers that are greater than x.

00:00:41.766 --> 00:00:46.766
So then we go to 15, then we go to 18,

00:00:50.658 --> 00:00:53.935
and then we go to 21.

00:00:53.935 --> 00:00:55.758
So we've now put them all in order.

00:00:55.758 --> 00:00:58.117
I knew I could put x here
because x is the median.

00:00:58.117 --> 00:00:58.628
When you put it in order,

00:00:58.628 --> 00:00:59.526
it's the middle number.

00:00:59.526 --> 00:01:01.268
Notice there's three
numbers larger than it,

00:01:01.268 --> 00:01:02.477
three numbers less than it.

00:01:02.477 --> 00:01:03.871
So I've put x in the middle,

00:01:03.871 --> 00:01:05.520
and so x has to be some number

00:01:05.520 --> 00:01:06.634
between 10 and 15,

00:01:06.634 --> 00:01:09.563
and it could be 10 or 15.

00:01:09.563 --> 00:01:10.736
But let's see the choices.

00:01:10.736 --> 00:01:12.860
Which of these choices
are between 10 and 15?

00:01:12.860 --> 00:01:15.125
Well five's not, eight's not, nine's not,

00:01:15.125 --> 00:01:16.598
14 is, we're done.

00:01:16.598 --> 00:01:18.723
16 is not, it's larger than 15,

00:01:18.723 --> 00:01:19.581
so we're done.

00:01:19.581 --> 00:01:22.577
X could be 14, it could also be 13,

00:01:22.577 --> 00:01:24.121
it could be 11, it could even be 10,

00:01:24.121 --> 00:01:26.762
it could be 15, but out
of all of the choices,

00:01:26.762 --> 00:01:30.200
14 is the only one that
meets our constraints.


2 Solving for mk

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00:00:00.838 --> 00:00:03.738
If two m times k is equal to 6.

00:00:03.757 --> 00:00:06.317
Them m times k is equal to.

00:00:06.317 --> 00:00:08.914
So let's rewrite that down here.

00:00:08.914 --> 00:00:11.225
Two m times k is equal to 6.

00:00:11.225 --> 00:00:13.976
And we need to figure out what m times k is.

00:00:13.983 --> 00:00:17.377
Well we can really just put these parenthesis around mk.

00:00:17.377 --> 00:00:23.294
Multiplication is associative. So this can be rewritten as two times mk is equal to 6.

00:00:23.294 --> 00:00:26.335
And that becomes pretty obvious how to solve for mk.

00:00:26.335 --> 00:00:29.764
Divide both sides by two you're just going to have mk on the left hand side.

00:00:29.764 --> 00:00:30.841
So let's do that.

00:00:30.841 --> 00:00:32.662
Divide both sides by 2.

00:00:32.662 --> 00:00:39.452
You're left with m times k is equal to six over two. Or three.