Example: 2-digit times 2-digit

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:04.580
Compute 23 times 44.

00:00:04.580 --> 00:00:06.550
And maybe the hardest part of
this problem, or maybe the

00:00:06.550 --> 00:00:09.470
first hard part, is to recognize
that that dot even

00:00:09.470 --> 00:00:10.750
means multiplication.

00:00:10.750 --> 00:00:15.700
This could have also been
written as 23 times 44, or

00:00:15.700 --> 00:00:20.880
they could have written it as 23
in parentheses times 44, so

00:00:20.880 --> 00:00:22.130
you just put the
two parentheses

00:00:22.130 --> 00:00:22.600
next to each other.

00:00:22.600 --> 00:00:24.840
That also implies
multiplication.

00:00:24.840 --> 00:00:27.860
So now that we know we're
multiplying, let's actually do

00:00:27.860 --> 00:00:28.570
the problem.

00:00:28.570 --> 00:00:32.290
So we're going to multiply 23--
I'll write it bigger.

00:00:32.290 --> 00:00:36.060
We're going to multiply
23 by 44.

00:00:36.060 --> 00:00:38.370
I'll write the traditional
multiplication sign there,

00:00:38.370 --> 00:00:40.740
just so that we know
we're multiplying.

00:00:40.740 --> 00:00:43.900
When you write it vertically
like this, you very seldom put

00:00:43.900 --> 00:00:44.695
a dot there.

00:00:44.695 --> 00:00:48.110
So let's do some
multiplication.

00:00:48.110 --> 00:00:50.610
Let's start off multiplying
this 4 in the

00:00:50.610 --> 00:00:53.310
ones place times 23.

00:00:53.310 --> 00:00:56.790
So you have 3 times 4 is 12.

00:00:56.790 --> 00:01:00.530
We can write 2 in the ones
place, but then we want to

00:01:00.530 --> 00:01:02.770
carry the 1, or we want
to regroup that

00:01:02.770 --> 00:01:04.379
1 in the tens place.

00:01:04.379 --> 00:01:07.310
So it's 12, so you put
the 1 over here.

00:01:07.310 --> 00:01:14.420
And now you have 4 times
2 is 8 plus 1 is 9.

00:01:14.420 --> 00:01:17.610
So you can think about
it as 4, this 4 right

00:01:17.610 --> 00:01:20.980
here, times 23 is 92.

00:01:20.980 --> 00:01:23.010
That's what we just
solved for.

00:01:23.010 --> 00:01:26.540
Now, we want to figure out
what this 4 times 23 is.

00:01:26.540 --> 00:01:29.420
Now what we do here is, when you
just do it mechanically,

00:01:29.420 --> 00:01:31.800
when you just learn the process,
you stick a 0 here.

00:01:31.800 --> 00:01:34.190
But the whole reason why you're
putting a 0 here is

00:01:34.190 --> 00:01:37.230
because you're now dealing with
a 4 in the tens place.

00:01:37.230 --> 00:01:39.750
If you had another-- I don't
know, a 3 or a 4 or whatever

00:01:39.750 --> 00:01:42.170
digit, and you're dealing with
the hundreds place, you'd put

00:01:42.170 --> 00:01:45.600
more zeroes here, because we're
going to find out 4

00:01:45.600 --> 00:01:46.900
times 23 is 92.

00:01:46.900 --> 00:01:47.970
We just figured that out.

00:01:47.970 --> 00:01:50.860
If we just multiplied this
4 times 23 again, we

00:01:50.860 --> 00:01:52.320
would get 92 again.

00:01:52.320 --> 00:01:56.320
But this 4 is actually a 40, so
it actually should be 920,

00:01:56.320 --> 00:01:57.840
and that's why we're
putting that 0.

00:01:57.840 --> 00:01:59.410
Now you're going to see
it in a second.

00:01:59.410 --> 00:02:01.930
So we have-- so let me put this
in a different color.

00:02:01.930 --> 00:02:04.410
So this 4 now we're
multiplying.

00:02:04.410 --> 00:02:06.740
4 times 3 is 12.

00:02:06.740 --> 00:02:09.150
Let's put the 2 right here.

00:02:09.150 --> 00:02:11.480
It should be in the tens place
because this is really a 40

00:02:11.480 --> 00:02:12.140
times the 3.

00:02:12.140 --> 00:02:13.360
Just think about it,
or you could just

00:02:13.360 --> 00:02:14.180
think of the process.

00:02:14.180 --> 00:02:15.890
It's the next space
that's free.

00:02:15.890 --> 00:02:17.440
4 times 3 is 12.

00:02:17.440 --> 00:02:18.650
Carry the 1.

00:02:18.650 --> 00:02:20.370
This blue 1 is from last time.

00:02:20.370 --> 00:02:21.400
You ignore it now.

00:02:21.400 --> 00:02:23.450
You don't want to make
that mess it up.

00:02:23.450 --> 00:02:25.190
That's when we multiplied
this 4.

00:02:25.190 --> 00:02:30.990
So now we have 4 times
2 is 8 plus 1 is 9.

00:02:30.990 --> 00:02:36.400
So what we figured out so far is
4 times 23 is 92, and this

00:02:36.400 --> 00:02:41.590
green 4 times 23 is 920, and
that's because this green 4

00:02:41.590 --> 00:02:42.920
actually represents 40.

00:02:42.920 --> 00:02:44.640
It's in the tens place.

00:02:44.640 --> 00:02:48.560
So when you multiply 44 times
23, it's going to be 4 times

00:02:48.560 --> 00:02:54.940
23, which is 92, plus 40
times 23, which is 920.

00:02:54.940 --> 00:02:56.300
I just want to make
sure we understand

00:02:56.300 --> 00:02:57.530
what we're doing here.

00:02:57.530 --> 00:02:59.770
And so we can take
their sum now.

00:02:59.770 --> 00:03:01.060
Let's add them up.

00:03:01.060 --> 00:03:03.120
2 plus 0 is 2.

00:03:03.120 --> 00:03:05.940
9 plus 2 is 11.

00:03:05.940 --> 00:03:07.090
Carry the 1.

00:03:07.090 --> 00:03:09.350
1 plus 9 is 10.

00:03:09.350 --> 00:03:10.910
Put a comma here, just
so it's easy to

00:03:10.910 --> 00:03:12.460
read, every third digit.

00:03:12.460 --> 00:03:15.820
So 23 times 44 is 1,012.


Adding Whole Numbers and Applications 4

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.880
Mike plans to build a porch
with the floor plan below.

00:00:03.880 --> 00:00:07.620
In order for the porch to be
up to city code, it cannot

00:00:07.620 --> 00:00:10.670
have a perimeter that is
greater than 300 feet.

00:00:10.670 --> 00:00:11.510
Let me underline that.

00:00:11.510 --> 00:00:15.840
It cannot have a perimeter that
is greater than 300 feet.

00:00:15.840 --> 00:00:18.050
Or another way to think about
it, the perimeter has to be

00:00:18.050 --> 00:00:20.290
300 feet or less.

00:00:20.290 --> 00:00:23.200
Find the perimeter of the porch
using the floor plan

00:00:23.200 --> 00:00:26.160
below and determine whether or
not its perimeter meets the

00:00:26.160 --> 00:00:28.010
city's code requirements.

00:00:28.010 --> 00:00:30.665
All right, so we need to find
the perimeter of this porch,

00:00:30.665 --> 00:00:33.260
and just as a reminder, the
perimeter is the distance

00:00:33.260 --> 00:00:34.570
around the object.

00:00:34.570 --> 00:00:36.550
Or another way of thinking about
it, it's the sum of the

00:00:36.550 --> 00:00:38.630
lengths of the sides
of the object.

00:00:38.630 --> 00:00:41.150
So to find the perimeter of this
porch, we just have to

00:00:41.150 --> 00:00:43.810
add up these lengths of
the different sides.

00:00:43.810 --> 00:00:44.700
So let's do that.

00:00:44.700 --> 00:00:54.960
So we're going to add up 40 plus
20 plus 55 plus 30 plus

00:00:54.960 --> 00:01:01.950
95 plus 50, and we're adding,
just to make sure we know what

00:01:01.950 --> 00:01:03.130
we're doing.

00:01:03.130 --> 00:01:05.040
We make sure we've aligned
them by place,

00:01:05.040 --> 00:01:05.890
which we've done here.

00:01:05.890 --> 00:01:07.370
The ones places are
all aligned.

00:01:07.370 --> 00:01:09.040
The tens places are
all aligned.

00:01:09.040 --> 00:01:10.820
So now let's start on
the ones place.

00:01:10.820 --> 00:01:15.830
What is 0 plus 0 plus 5
plus 0 plus 5 plus 0?

00:01:15.830 --> 00:01:18.600
Well, that's really just the
same thing as 5 plus 5, which

00:01:18.600 --> 00:01:19.980
is 10, right?

00:01:19.980 --> 00:01:22.960
The 0, carry the 1, or another
way of thinking about it,

00:01:22.960 --> 00:01:25.640
regroup the 1, which is
really a 10, right?

00:01:25.640 --> 00:01:27.390
It's 10.

00:01:27.390 --> 00:01:31.630
The 1 in the tens place
literally represents 10, so we

00:01:31.630 --> 00:01:33.550
put it in the tens place,
and now we add up

00:01:33.550 --> 00:01:35.270
all of the tens places.

00:01:35.270 --> 00:01:42.690
So we have 1 plus 4 is 5, plus
2 is 7, plus 5 is 12, plus 3

00:01:42.690 --> 00:01:48.770
is 15, plus 9 is 24,
plus 5 is 29.

00:01:48.770 --> 00:01:54.280
So the total perimeter of Mike's
porch here is 290 feet,

00:01:54.280 --> 00:02:00.940
which is less than 300 feet,
so it meets the city code.

00:02:00.940 --> 00:02:03.840
It could have even been equal to
300 feet, so we could have

00:02:03.840 --> 00:02:06.880
even said it just has to be
less than or equal to 300

00:02:06.880 --> 00:02:08.280
feet, and we meet
our city code.

00:02:08.280 --> 00:02:11.419
So he can go ahead and
build this porch.


Adding Whole Numbers and Applications 3

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.830
Find the perimeter
of this hexagon.

00:00:03.830 --> 00:00:06.660
So the perimeter just means
the distance around the

00:00:06.660 --> 00:00:09.860
object, or if you add up all of
the lengths of the sides of

00:00:09.860 --> 00:00:11.970
the object, so we're going to
add up all of the lengths of

00:00:11.970 --> 00:00:15.530
sides of this hexagon, and a
hexagon is just a six-sided

00:00:15.530 --> 00:00:17.800
geometrical shape like this.

00:00:17.800 --> 00:00:20.180
So to find the perimeter, we
just need to find the distance

00:00:20.180 --> 00:00:23.650
around the hexagon, which is
going to be the sum of all of

00:00:23.650 --> 00:00:24.930
the lengths of the sides.

00:00:24.930 --> 00:00:27.800
So the perimeter is going to be
24-- so how many of these

00:00:27.800 --> 00:00:27.950
do we have?

00:00:27.950 --> 00:00:31.670
We have one, two, three, four,
five, six 24's, which makes

00:00:31.670 --> 00:00:34.170
sense, because a hexagon
has six sides.

00:00:34.170 --> 00:00:36.400
So we're going to have-- let
me write this a little bit

00:00:36.400 --> 00:00:51.320
bigger-- 24 plus 24 plus 24 plus
24 plus 24-- and that's

00:00:51.320 --> 00:00:54.300
five 24's-- plus 24.

00:00:54.300 --> 00:00:56.450
So let's add them all
at the same time.

00:00:56.450 --> 00:00:58.820
We could've done it piece by
piece, but let's just go

00:00:58.820 --> 00:00:59.490
straight ahead.

00:00:59.490 --> 00:01:01.680
So let's go first to
the ones place.

00:01:01.680 --> 00:01:12.360
So 4 plus 4 is 8, 8 plus 4 is
12, 12 plus 4 is 16, 16 plus 4

00:01:12.360 --> 00:01:20.060
is 20, 20 plus 4 is 24.

00:01:20.060 --> 00:01:22.050
And now we're in
the tens place.

00:01:22.050 --> 00:01:30.830
2 plus 2 is 4 plus 2 is 6 plus
2 is 8 plus 2 is 10 plus 2 is

00:01:30.830 --> 00:01:36.480
12 plus 2 is 14.

00:01:36.480 --> 00:01:41.650
So the sum of these
six 24's is 144.

00:01:41.650 --> 00:01:43.310
That is the perimeter.

00:01:43.310 --> 00:01:45.130
And when we learn
multiplication, if you haven't

00:01:45.130 --> 00:01:47.020
learned it already, you could've
down this simpler.

00:01:47.020 --> 00:01:49.380
You just say you have six 24's,
so this would actually

00:01:49.380 --> 00:01:52.850
be a multiplication problem
of 6 times 24.

00:01:52.850 --> 00:01:54.340
If you don't know how
to do that just yet,

00:01:54.340 --> 00:01:55.390
don't worry about it.

00:01:55.390 --> 00:01:56.690
You can just add them all up.

00:01:56.690 --> 00:01:58.200
You get 144.


Rounding to Estimate Differences

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:06.560
A microbiologist has 1,256
microbes growing in culture.

00:00:06.560 --> 00:00:09.880
When he checks again after
fifteen minutes, there are

00:00:09.880 --> 00:00:15.100
2,283 microbes in the same
culture, so they're growing

00:00:15.100 --> 00:00:18.790
pretty fast. Estimate the
difference between the number

00:00:18.790 --> 00:00:22.720
of microbes at each check by
rounding each number to the

00:00:22.720 --> 00:00:24.330
nearest ten.

00:00:24.330 --> 00:00:26.830
So what they want us to do,
they want us to round each

00:00:26.830 --> 00:00:31.470
number to the nearest ten, and
then with the rounded numbers,

00:00:31.470 --> 00:00:33.890
find the difference between
the two checks.

00:00:33.890 --> 00:00:35.940
So let's round each
of these numbers.

00:00:35.940 --> 00:00:44.910
So we started with
1,256 microbes.

00:00:44.910 --> 00:00:48.000
So how do we round this
to the nearest ten?

00:00:48.000 --> 00:00:49.590
That's what they
want us to do.

00:00:49.590 --> 00:00:51.140
So you look at the tens place.

00:00:51.140 --> 00:00:53.750
The tens place is that
5 right there.

00:00:53.750 --> 00:00:56.390
So if we round it up,
it would go to a 6.

00:00:56.390 --> 00:00:59.230
If we round it down, it
would stay at a 5.

00:00:59.230 --> 00:01:00.970
If we round it up,
this would be 60.

00:01:00.970 --> 00:01:03.400
If we round it down,
this will be 50.

00:01:03.400 --> 00:01:06.730
And the rule for rounding is
you look at the place one

00:01:06.730 --> 00:01:07.310
below that.

00:01:07.310 --> 00:01:10.680
You go to the ones place in this
case, and you see if that

00:01:10.680 --> 00:01:13.990
is 5 or more, or 5 or
greater, and it is.

00:01:13.990 --> 00:01:16.420
It's 6, which is definitely
5 or greater, so we

00:01:16.420 --> 00:01:18.270
want to round up.

00:01:18.270 --> 00:01:25.200
So we want to round up, and
we're rounding up to the

00:01:25.200 --> 00:01:25.950
nearest ten.

00:01:25.950 --> 00:01:34.690
So this will be go from 56
to 60, so it'll be 1,260.

00:01:34.690 --> 00:01:36.380
We've rounded up.

00:01:36.380 --> 00:01:39.720
Now, when we wait fifteen
minutes, we check and we see

00:01:39.720 --> 00:01:47.640
that there are 2,283 microbes.

00:01:47.640 --> 00:01:51.400
Once again, let's look
at the tens place.

00:01:51.400 --> 00:01:52.600
That's the 8 right there.

00:01:52.600 --> 00:01:54.250
Let me do that in a
different color.

00:01:54.250 --> 00:01:56.580
That is the 8 right there.

00:01:56.580 --> 00:01:58.380
And you look at the place
one below that.

00:01:58.380 --> 00:02:00.860
If that's 5 or greater,
you round up.

00:02:00.860 --> 00:02:02.480
Otherwise, you round down.

00:02:02.480 --> 00:02:05.170
This right here, 3 is
less than 5 so you

00:02:05.170 --> 00:02:08.919
want to round down.

00:02:08.919 --> 00:02:12.350
We want to round down, so we're
going to be left with

00:02:12.350 --> 00:02:17.840
2,200, and instead of 83, we're
just going to have 80.

00:02:17.840 --> 00:02:21.010
2,280.

00:02:21.010 --> 00:02:23.800
Now, they want us to estimate
the difference between the

00:02:23.800 --> 00:02:26.830
number of microbes at each check
by rounding each number

00:02:26.830 --> 00:02:29.070
to the nearest ten, so
we've done that part.

00:02:29.070 --> 00:02:30.640
We've rounded each number
to the nearest ten.

00:02:30.640 --> 00:02:32.330
Now we have to find
the difference.

00:02:32.330 --> 00:02:35.460
So let's subtract the first
check from the second check.

00:02:35.460 --> 00:02:36.980
That's how we get a
positive number.

00:02:36.980 --> 00:02:38.080
So let's do that.

00:02:38.080 --> 00:02:43.030
So if we subtract this 1,260--
let me just copy and paste it.

00:02:43.030 --> 00:02:48.280
So we want to subtract this--
copy it and paste it.

00:02:48.280 --> 00:02:52.730
We want to subtract that from
the 2,280 to figure out the

00:02:52.730 --> 00:02:53.900
difference.

00:02:53.900 --> 00:02:55.270
So we're going to
subtract here.

00:02:55.270 --> 00:02:58.020
Let me put the subtraction
sign, and let's do some

00:02:58.020 --> 00:02:58.730
subtraction.

00:02:58.730 --> 00:03:00.900
So you go first to
the ones place.

00:03:00.900 --> 00:03:02.270
0 minus 0.

00:03:02.270 --> 00:03:04.590
Well, that's just
going to be 0.

00:03:04.590 --> 00:03:08.100
8 minus 6 in the tens place,
it's really 80 minus 60.

00:03:08.100 --> 00:03:10.035
That's going to be 2, but since
it's in the tens place,

00:03:10.035 --> 00:03:12.960
it's 20, or you could just
do 80 minus 60 is 20, so

00:03:12.960 --> 00:03:14.640
everything makes sense so far.

00:03:14.640 --> 00:03:20.010
2 minus 2 is 0, and then
2 minus 1 is 1.

00:03:20.010 --> 00:03:23.010
So when we rounded each number
to the nearest ten and then

00:03:23.010 --> 00:03:26.780
took the difference, our
estimated difference is 1,020.


Rounding to Estimate Sums

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:10.380
Estimate the solution to 582,205
plus 610,859, rounding

00:00:10.380 --> 00:00:13.660
these two numbers to the
nearest thousand.

00:00:13.660 --> 00:00:15.430
So we going to round each of
these numbers to the nearest

00:00:15.430 --> 00:00:17.980
thousand and then add them,
and that'll give us an

00:00:17.980 --> 00:00:19.440
estimate to this solution.

00:00:19.440 --> 00:00:21.310
So let's do each of
these numbers.

00:00:21.310 --> 00:00:31.760
So if we start off with 582,205,
we want to round to

00:00:31.760 --> 00:00:32.795
the nearest thousand.

00:00:32.795 --> 00:00:35.860
So you go to the thousands
place, and you go one place

00:00:35.860 --> 00:00:36.740
below that.

00:00:36.740 --> 00:00:39.310
If that's 5 or greater,
round up.

00:00:39.310 --> 00:00:41.240
If it's less than
5, round down.

00:00:41.240 --> 00:00:42.620
This is less than 5.

00:00:42.620 --> 00:00:47.010
The 2 here is less than 5,
so we want round down.

00:00:47.010 --> 00:00:52.860
We are going to round it down,
so it's going to be 582,000.

00:00:52.860 --> 00:00:54.560
We're just going to round
it down to 2,000.

00:00:54.560 --> 00:00:57.100
We're going to get rid of
all of this 205 here.

00:00:57.100 --> 00:01:02.910
582,000 when we rounded it down
to the nearest thousand.

00:01:02.910 --> 00:01:10.910
Now, we have 610,859.

00:01:10.910 --> 00:01:12.090
Same drill.

00:01:12.090 --> 00:01:13.670
Go to the thousands place.

00:01:13.670 --> 00:01:15.560
That's the 0 right there.

00:01:15.560 --> 00:01:18.300
Look one place below the
thousands place.

00:01:18.300 --> 00:01:20.080
This is that 8 right there.

00:01:20.080 --> 00:01:22.850
If this is 5 or greater,
you round up.

00:01:22.850 --> 00:01:24.310
Less than 5, round down.

00:01:24.310 --> 00:01:25.460
It is 5 or greater.

00:01:25.460 --> 00:01:27.160
8 is greater than 5.

00:01:27.160 --> 00:01:29.950
Or it could even be 5 and we'd
still around up, so we're

00:01:29.950 --> 00:01:32.240
going to round up in
this situation.

00:01:32.240 --> 00:01:36.680
We're going to round up, so we
end up with 600, and instead

00:01:36.680 --> 00:01:43.570
of 610,000, we're rounding
up to 611,000.

00:01:43.570 --> 00:01:45.790
We've rounded the two numbers
to the nearest thousand.

00:01:45.790 --> 00:01:46.796
Now we're ready to add.

00:01:46.796 --> 00:01:49.490
So let's add them.

00:01:49.490 --> 00:01:51.330
Let's add the two numbers.

00:01:51.330 --> 00:01:52.800
0 plus 0 is 0.

00:01:52.800 --> 00:01:54.400
0 plus 0 is 0.

00:01:54.400 --> 00:01:56.860
0 plus 0 is 0.

00:01:56.860 --> 00:02:00.230
2 plus 1 is 3.

00:02:00.230 --> 00:02:02.440
8 plus 1 is 9.

00:02:02.440 --> 00:02:04.230
5 plus 6 is 11.

00:02:04.230 --> 00:02:09.130
If there were more places here,
we would carry the 1 in

00:02:09.130 --> 00:02:10.030
the tens place.

00:02:10.030 --> 00:02:13.230
But there isn't, so you can
just write an 11 here.

00:02:13.230 --> 00:02:16.380
And so we are left with-- put
a comma every third digit--

00:02:16.380 --> 00:02:20.200
1,193,000, which is
our estimate.

00:02:20.200 --> 00:02:22.790
And the whole reason why this
is useful is if someone were

00:02:22.790 --> 00:02:24.740
to give these numbers to you and
you had to do it in your

00:02:24.740 --> 00:02:28.080
head, it's hard to add these
whole numbers, but you might

00:02:28.080 --> 00:02:31.240
be able to do 582 plus
611, knowing

00:02:31.240 --> 00:02:32.830
everything isn't thousands.

00:02:32.830 --> 00:02:34.780
So it allows you to do things
in your head a little bit

00:02:34.780 --> 00:02:38.550
easier when you round, and then
you estimate the sum of

00:02:38.550 --> 00:02:40.570
the rounded numbers.


Rounding to Estimate Sums 2

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:06.520
A town had a population
of 11,256 in 1995.

00:00:06.520 --> 00:00:09.340
By 2010, the population
in that town had

00:00:09.340 --> 00:00:12.560
increased by 4,312.

00:00:12.560 --> 00:00:16.290
Estimate the town's population
in 2010 by first rounding

00:00:16.290 --> 00:00:18.300
these two numbers to the
nearest hundred.

00:00:18.300 --> 00:00:20.470
So we could figure out the
exact population just by

00:00:20.470 --> 00:00:22.400
adding these two, but they
want us to round to the

00:00:22.400 --> 00:00:24.930
nearest hundred first and get an
estimate, and then add, so

00:00:24.930 --> 00:00:27.430
it'll be an estimate of
the town's population.

00:00:27.430 --> 00:00:30.840
So first we have this 11,256.

00:00:30.840 --> 00:00:33.130
That's the population in 1995.

00:00:33.130 --> 00:00:35.610
11,256.

00:00:35.610 --> 00:00:38.020
We need to round to the nearest
hundred, so let's look

00:00:38.020 --> 00:00:39.350
at the hundreds place here.

00:00:39.350 --> 00:00:43.790
So this is either going to go
down to 200 or go up to 300,

00:00:43.790 --> 00:00:46.130
depending on what the
next place is or

00:00:46.130 --> 00:00:47.290
the next lowest place.

00:00:47.290 --> 00:00:50.370
So we go down to the tens
place, we see a 5.

00:00:50.370 --> 00:00:55.140
5 is 5 or greater, so we want
to round this number up.

00:00:55.140 --> 00:00:57.000
We are going to round up.

00:00:57.000 --> 00:00:58.220
So if you round up
to the nearest

00:00:58.220 --> 00:00:59.960
hundred, you get 11,000.

00:00:59.960 --> 00:01:03.970
Instead of having 256, we're
rounding up to 300.

00:01:03.970 --> 00:01:08.170
300 is closer to 256
than 200 is.

00:01:08.170 --> 00:01:09.770
That's why we're rounding
up to that.

00:01:09.770 --> 00:01:12.010
That's why it's to the
nearest hundred.

00:01:12.010 --> 00:01:19.780
Now, we added 4,312
people, so 4,312.

00:01:19.780 --> 00:01:22.270
Now once again, we look
at the hundreds place.

00:01:22.270 --> 00:01:23.920
We go one spot below that.

00:01:23.920 --> 00:01:26.730
The question we're trying to
answer: Do we go down to 4,300

00:01:26.730 --> 00:01:28.900
or do we go up to 4,400?

00:01:28.900 --> 00:01:30.370
Which one is closer?

00:01:30.370 --> 00:01:32.230
Well, we see we only
have a 1 here.

00:01:32.230 --> 00:01:38.420
1 is definitely less than 5,
so we want to round down.

00:01:38.420 --> 00:01:40.930
So 312 is closer to 300.

00:01:40.930 --> 00:01:43.850
We round down to just a 3 here
and these get zeroed out.

00:01:43.850 --> 00:01:47.310
So 4,300.

00:01:47.310 --> 00:01:51.520
And now we're ready to add
our rounded numbers.

00:01:51.520 --> 00:01:53.700
0 plus 0 is 0.

00:01:53.700 --> 00:01:56.160
0 plus 0 is 0.

00:01:56.160 --> 00:01:58.140
3 plus 3 is 6.

00:01:58.140 --> 00:01:59.630
Let's put a comma here
so it's easy to

00:01:59.630 --> 00:02:00.880
read every third digit.

00:02:00.880 --> 00:02:05.080
1 plus 4 is 5, and then 1 plus--
we have nothing here,

00:02:05.080 --> 00:02:06.480
so we just bring down that 1.

00:02:06.480 --> 00:02:07.710
We could have written
that addition there.

00:02:07.710 --> 00:02:08.944
We're adding those
two numbers.

00:02:08.944 --> 00:02:14.005
So our estimate of the town's
population in 2010 is 15,600.


Example: Ways to represent multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:06.410
Rewrite 5 plus 5 plus 5 plus
5 plus 5 plus 5 plus 5 as a

00:00:06.410 --> 00:00:08.440
multiplication expression.

00:00:08.440 --> 00:00:11.120
And then they want us to write
the expression three times

00:00:11.120 --> 00:00:14.260
using different ways to
write multiplication.

00:00:14.260 --> 00:00:15.610
So let's do the first part.

00:00:15.610 --> 00:00:17.390
Let's write it as a
multiplication expression.

00:00:17.390 --> 00:00:19.945
So how many times have
we added 5 here?

00:00:19.945 --> 00:00:23.660
Well, we've got it at
one, two, three,

00:00:23.660 --> 00:00:26.670
four, five, six, seven.

00:00:26.670 --> 00:00:29.110
So one way to think of it, if
I just said what is here?

00:00:29.110 --> 00:00:30.240
How many 5's are there?

00:00:30.240 --> 00:00:34.330
You'd say, well, I added 5 to
itself seven times, right?

00:00:34.330 --> 00:00:38.360
You could literally say
this is 7 times 5.

00:00:38.360 --> 00:00:44.350
We could literally write, this
is 7 times 5, or you could

00:00:44.350 --> 00:00:47.230
view it as 5 seven times.

00:00:47.230 --> 00:00:48.990
I'm not even writing it
mathematically yet.

00:00:48.990 --> 00:00:51.470
I'm just saying, look, if I saw
seven of something, you

00:00:51.470 --> 00:00:54.010
would literally say, if these
were apples, you would say

00:00:54.010 --> 00:00:58.030
apples seven times, or you'd
say seven times the apple,

00:00:58.030 --> 00:00:59.040
whatever it is.

00:00:59.040 --> 00:01:00.810
Now, in this case, we're
actually adding the number to

00:01:00.810 --> 00:01:03.180
each other, and we could figure
out what that is, and

00:01:03.180 --> 00:01:04.160
why don't we?

00:01:04.160 --> 00:01:06.680
But the way we would write this
mathematically, we would

00:01:06.680 --> 00:01:10.090
say this is 7 times 5.

00:01:10.090 --> 00:01:11.610
We could also write
it like this.

00:01:11.610 --> 00:01:15.190
We could write it 7 dot 5.

00:01:15.190 --> 00:01:17.580
This and this mean the
exact same thing.

00:01:17.580 --> 00:01:21.440
It means we're multiplying
7 times 5 or 5 times 7.

00:01:21.440 --> 00:01:24.110
You can actually switch the
order, and you get the exact

00:01:24.110 --> 00:01:25.250
same value.

00:01:25.250 --> 00:01:27.890
You could actually write
it 5 times 7.

00:01:27.890 --> 00:01:32.160
So you could interpret this as 7
five times or 5 seven times,

00:01:32.160 --> 00:01:34.840
however you like to do
it, or 5 seven times.

00:01:34.840 --> 00:01:35.610
I don't want to confuse you.

00:01:35.610 --> 00:01:37.350
I just want to show you that
these are all equivalent.

00:01:37.350 --> 00:01:39.250
This is also equivalent.

00:01:39.250 --> 00:01:41.150
5 times 7.

00:01:41.150 --> 00:01:42.080
Same thing.

00:01:42.080 --> 00:01:44.050
You could write them
in parentheses.

00:01:44.050 --> 00:01:45.560
You could write it like this.

00:01:45.560 --> 00:01:47.300
This all means the same thing.

00:01:47.300 --> 00:01:50.040
That's 7 times 5,
and so is this.

00:01:50.040 --> 00:01:53.220
These all evaluate to the
same thing: 5 times 7.

00:01:53.220 --> 00:01:56.030
So these are all equivalent, and
since we've worked with it

00:01:56.030 --> 00:01:57.720
so much, let's just figure
out the answer.

00:01:57.720 --> 00:02:00.880
So if we add up 5 to itself
seven times, what do we get?

00:02:00.880 --> 00:02:02.740
Well, 5 plus 5 is 10.

00:02:02.740 --> 00:02:09.210
10 plus 5 is 15, plus 5 is 20,
plus 5 is 25, plus 5 is 30,

00:02:09.210 --> 00:02:10.789
plus 5 is 35.

00:02:10.789 --> 00:02:22.150
So all of these evaluate to
35, just so you see that

00:02:22.150 --> 00:02:23.550
they're the same thing.

00:02:23.550 --> 00:02:26.790
These are all equivalent
to 35.

00:02:26.790 --> 00:02:29.850
And just something to think
about, this is also the exact

00:02:29.850 --> 00:02:32.350
same thing, depending on how you
want to interpret this, as

00:02:32.350 --> 00:02:33.695
7 five times.

00:02:36.660 --> 00:02:38.510
They didn't ask us to do it, but
I thought I would point it

00:02:38.510 --> 00:02:39.380
out to you.

00:02:39.380 --> 00:02:46.080
7 five times would look like
this: 7 plus 7 plus 7 plus 7

00:02:46.080 --> 00:02:47.410
plus 7, right?

00:02:47.410 --> 00:02:49.950
I have 7 five times.

00:02:49.950 --> 00:02:52.370
I added it to itself five
different times.

00:02:52.370 --> 00:02:54.820
There's five 7's here
added to each other.

00:02:54.820 --> 00:02:57.050
And when you add these up,
you'll also get 35.

00:02:57.050 --> 00:03:02.560
And that's why 5 times 7 and 7
times 5 is the same thing.


Multiplying Whole Numbers and Applications 3

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:07.620
Compute twenty-five hundred
times 80, or 2,500 times 80.

00:00:07.620 --> 00:00:10.160
Now, we could just do this
multiplication problem like a

00:00:10.160 --> 00:00:12.725
traditional one, but when you
have these numbers with a lot

00:00:12.725 --> 00:00:15.340
of zeroes there, and it actually
sometimes helps you

00:00:15.340 --> 00:00:17.010
do it in your head, is
to kind of think

00:00:17.010 --> 00:00:18.260
about the zeroes later.

00:00:18.260 --> 00:00:20.120
And the reason why we can do
that-- let's rewrite this

00:00:20.120 --> 00:00:21.580
multiplication problem.

00:00:21.580 --> 00:00:25.920
2,500 times 80.

00:00:25.920 --> 00:00:29.020
This is equivalent to--
2,500 is what?

00:00:29.020 --> 00:00:30.760
It's 25 times 100.

00:00:30.760 --> 00:00:34.680
This is equivalent
to 25 times 100.

00:00:34.680 --> 00:00:36.680
That's what 2,500 is.

00:00:36.680 --> 00:00:40.320
2,500 is 25 times
100 times 80.

00:00:40.320 --> 00:00:41.760
And what's 80?

00:00:41.760 --> 00:00:47.260
80 is 8 times 10.

00:00:47.260 --> 00:00:49.990
And in multiplication, you can
rearrange it any way you like.

00:00:49.990 --> 00:00:53.250
You can change the order when
it's all a bunch of numbers

00:00:53.250 --> 00:00:54.290
being multiplied.

00:00:54.290 --> 00:00:59.440
So this is the same thing, this
is the equivalent to 25

00:00:59.440 --> 00:01:10.610
times 8 times 100 times 10.

00:01:10.610 --> 00:01:12.130
And in the future, you're going
to be able to do this in

00:01:12.130 --> 00:01:14.950
your head, but I really want
to show you why you can do

00:01:14.950 --> 00:01:19.010
this in your head, why you can
just multiply 25 times 8, and

00:01:19.010 --> 00:01:20.920
then worry about the zeroes.

00:01:20.920 --> 00:01:22.570
So what's this going
to be equal to?

00:01:22.570 --> 00:01:28.210
So this is going to be
equal to 25 times 8.

00:01:28.210 --> 00:01:30.860
And what's 100 times 10?

00:01:30.860 --> 00:01:31.890
Well, 100 times 10 is 1,000.

00:01:31.890 --> 00:01:33.680
You can maybe do that
in your head.

00:01:36.200 --> 00:01:38.950
And whenever you multiply powers
of ten, the way you can

00:01:38.950 --> 00:01:40.720
think about it is you just
literally add the zeroes.

00:01:40.720 --> 00:01:42.290
You had two zeroes here.

00:01:42.290 --> 00:01:43.490
You have one zero here.

00:01:43.490 --> 00:01:46.040
When you take their product,
you're going to get something

00:01:46.040 --> 00:01:46.970
with three zeroes.

00:01:46.970 --> 00:01:50.180
100 times 10 is 1,000.

00:01:50.180 --> 00:01:52.870
And when we multiply this times
whatever this is, we'll

00:01:52.870 --> 00:01:55.430
literally just add three zeroes
to the product, so

00:01:55.430 --> 00:01:56.960
that's why this is useful.

00:01:56.960 --> 00:02:00.490
We can now think about this as
being-- this is the same thing

00:02:00.490 --> 00:02:06.540
as 25 times 8, and then whatever
25 times 8 is, we can

00:02:06.540 --> 00:02:12.460
then add three zeroes to it.

00:02:12.460 --> 00:02:14.780
Hopefully, this explained
a little bit about why.

00:02:14.780 --> 00:02:18.050
Let's figure out what
25 times 8 is.

00:02:18.050 --> 00:02:22.870
25 times 8.

00:02:22.870 --> 00:02:26.210
5 times 8 is 40.

00:02:26.210 --> 00:02:27.150
Put the zero down.

00:02:27.150 --> 00:02:29.400
Carry the 4.

00:02:29.400 --> 00:02:34.150
8 times 2 is 16 plus 4 is 20.

00:02:34.150 --> 00:02:37.380
So it's an even 200,
which makes sense.

00:02:37.380 --> 00:02:41.630
4 times 24 is 100, or four
quarters make a $1.00, so

00:02:41.630 --> 00:02:43.080
eight quarters would
make $2.00, so

00:02:43.080 --> 00:02:44.150
that all makes sense.

00:02:44.150 --> 00:02:47.440
So this part right here is
200, so we're going to

00:02:47.440 --> 00:02:52.650
multiply 200 times 1,000.

00:02:52.650 --> 00:02:55.980
The easiest way to think about
it is you just add the zeroes

00:02:55.980 --> 00:02:58.300
when you're multiplying
times a power of ten.

00:02:58.300 --> 00:03:01.580
So this is going to be-- we're
going to have a 200, or you

00:03:01.580 --> 00:03:04.510
could just imagine you have 2,
0, 0, and then you add the

00:03:04.510 --> 00:03:06.280
three zeroes there.

00:03:06.280 --> 00:03:11.550
One, two, three, and you
are left with 200,000.

00:03:11.550 --> 00:03:14.450
So 2,500 times 80: 200,000.

00:03:14.450 --> 00:03:16.400
You could've multiplied it out
in the traditional sense, but

00:03:16.400 --> 00:03:18.430
it's good to see this-- I guess
you could call it a

00:03:18.430 --> 00:03:20.490
trick or a way to do it in
your head a little bit.

00:03:20.490 --> 00:03:21.930
And you'll see that more
and more, the more

00:03:21.930 --> 00:03:23.710
practice you get.


Multiplication estimation example

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.840
A ticket agent sells 42
tickets to a play.

00:00:03.840 --> 00:00:06.240
The tickets cost $29 each.

00:00:06.240 --> 00:00:10.600
Use rounding to estimate the
total dollars taken in from

00:00:10.600 --> 00:00:12.320
the sale of the tickets.

00:00:12.320 --> 00:00:16.950
Now if we wanted the exact
number, we could say 42 times

00:00:16.950 --> 00:00:19.710
29, and we could work out the
multiplication, but they

00:00:19.710 --> 00:00:21.430
essentially want us to be able
to do it in our head.

00:00:21.430 --> 00:00:24.310
We want to round the numbers
first and then multiply.

00:00:24.310 --> 00:00:27.160
So if we want to round, and
really we just have two places

00:00:27.160 --> 00:00:28.670
here, so if we're going to round
anything, it's going to

00:00:28.670 --> 00:00:31.130
be to the nearest ten because
that's the largest place we

00:00:31.130 --> 00:00:34.670
have. So if we round 42 to the
nearest ten-- we've done this

00:00:34.670 --> 00:00:38.680
drill many times-- 2 in the ones
place is the less than 5,

00:00:38.680 --> 00:00:40.030
so we're going to round down.

00:00:40.030 --> 00:00:41.540
The nearest ten is 40.

00:00:41.540 --> 00:00:44.050
We're going to round
down to 40.

00:00:44.050 --> 00:00:48.940
29, if we round to the nearest
ten, 9 in the ones place is

00:00:48.940 --> 00:00:54.670
greater than or equal to
5, so we round up.

00:00:54.670 --> 00:00:57.100
The nearest ten is 30.

00:00:57.100 --> 00:00:58.240
And another way to
think about it.

00:00:58.240 --> 00:01:00.990
Just say, well, you know, 42,
that's pretty close to 40.

00:01:00.990 --> 00:01:02.870
29 is pretty close to 30.

00:01:02.870 --> 00:01:05.450
Those are literally the nearest
multiples of ten that

00:01:05.450 --> 00:01:07.520
I can figure out, so
now I can multiply.

00:01:10.180 --> 00:01:12.900
And here, once again, we can
use-- you could call it a

00:01:12.900 --> 00:01:15.540
trick, but hopefully, you
understand why it works.

00:01:15.540 --> 00:01:17.830
But 30 times 40, instead of
you saying, well, this is

00:01:17.830 --> 00:01:21.050
going to be the same thing as 3
times 4, but we're going to

00:01:21.050 --> 00:01:23.290
put two zeroes at
the end of it.

00:01:23.290 --> 00:01:26.180
30 times 40 is the same thing as
3 times 4 with two zeroes,

00:01:26.180 --> 00:01:26.860
so let's do that.

00:01:26.860 --> 00:01:32.650
So you have 3 times 4 is 12,
which we know, and then we

00:01:32.650 --> 00:01:34.340
have two zeroes.

00:01:34.340 --> 00:01:36.820
We got that zero, so let's stick
that zero there, and

00:01:36.820 --> 00:01:38.760
then we got that blue
zero there, so let's

00:01:38.760 --> 00:01:40.080
put that over there.

00:01:40.080 --> 00:01:46.080
So they're going to have roughly
$1,200 taken it from

00:01:46.080 --> 00:01:47.400
sales of the tickets.

00:01:47.400 --> 00:01:49.510
That is our estimate.


Multiplying Whole Numbers and Applications 5

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.400
Calculate the area of a
rectangular room that is 13

00:00:04.400 --> 00:00:08.000
feet wide and 18 feet long.

00:00:08.000 --> 00:00:09.860
So the room might
look like this.

00:00:09.860 --> 00:00:11.470
We're looking at a floor plan.

00:00:11.470 --> 00:00:20.080
So it is 13 feet wide and
it is 18 feet long.

00:00:20.080 --> 00:00:22.130
So that dimension is 18 feet.

00:00:22.130 --> 00:00:24.920
And when you're looking for the
area of a rectangle, you

00:00:24.920 --> 00:00:27.440
literally just multiply
the two dimensions.

00:00:27.440 --> 00:00:30.850
You multiply the width times
the length in this case.

00:00:30.850 --> 00:00:36.770
So the area is going
to be equal to 13

00:00:36.770 --> 00:00:39.265
times 18 square feet.

00:00:39.265 --> 00:00:40.750
So let's multiply that out.

00:00:40.750 --> 00:00:46.690
13 times 18.

00:00:46.690 --> 00:00:49.320
Now we're first going to worry
about this 8 in the ones

00:00:49.320 --> 00:00:49.980
place, right?

00:00:49.980 --> 00:00:53.000
18 is really 10 plus 8.

00:00:53.000 --> 00:00:57.820
So when we take this 8 in the
ones place, 8 times 3 is 24,

00:00:57.820 --> 00:00:59.400
the 4 in the ones places.

00:00:59.400 --> 00:01:03.230
Regroup the 2, which is really
a 20, or carry the 2.

00:01:03.230 --> 00:01:07.390
8 times 1, now this 1 is really
a 10, so 8 times 1 is

00:01:07.390 --> 00:01:11.510
8, plus 2 is 10, and there's
no other digits, so we can

00:01:11.510 --> 00:01:13.690
write the whole 10 down here.

00:01:13.690 --> 00:01:19.170
So we now see that 8
times 13 is 104.

00:01:19.170 --> 00:01:21.340
Now let's worry about
the next digit.

00:01:21.340 --> 00:01:24.950
Let us worry about this 1 right
here in the tens place.

00:01:24.950 --> 00:01:27.170
Now since it's in the tens
place, we want to

00:01:27.170 --> 00:01:29.250
put a zero down here.

00:01:29.250 --> 00:01:32.230
1 times 3 is 3.

00:01:32.230 --> 00:01:34.370
1 times 1, we ignore this 2.

00:01:34.370 --> 00:01:36.200
That was from the other
digit so we ignore it.

00:01:36.200 --> 00:01:38.120
1 times 1 is 1.

00:01:38.120 --> 00:01:43.400
So you see that 8 times 13 is
104, 1 times 13, or we should

00:01:43.400 --> 00:01:45.510
say 10 times 13, because
that 1 is in the

00:01:45.510 --> 00:01:47.740
tens place, is 130.

00:01:47.740 --> 00:01:52.200
18 is the sum of 10 and 8, so 18
times 13 is going to be the

00:01:52.200 --> 00:01:54.310
sum of these two numbers.

00:01:54.310 --> 00:01:57.260
So it's going to be
104 plus 130.

00:01:57.260 --> 00:02:03.310
4 plus 0 is 4, 0 plus 3
is 3, 1 plus 1 is 2.

00:02:03.310 --> 00:02:09.152
So the area of the room is 234
square feet, or you could call

00:02:09.152 --> 00:02:11.530
it feet squared, however
you want to do it.

00:02:11.530 --> 00:02:16.860
I'll write it like this:
square feet.

00:02:16.860 --> 00:02:20.260
And if you're actually draw a
grid where you drew 18 lines

00:02:20.260 --> 00:02:22.980
in this direction, if you kept
going, and 13 lines in this

00:02:22.980 --> 00:02:27.740
direction, if you were to
count all the boxes,

00:02:27.740 --> 00:02:29.820
you would get 234.


Finding a number's place value

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:06.730
Find the place value
of 3 in 4,356.

00:00:06.730 --> 00:00:09.190
Now, whenever I think about
place value, and the more you

00:00:09.190 --> 00:00:11.720
do practice problems on this
it'll become a little bit of

00:00:11.720 --> 00:00:14.600
second nature, but whenever I
see a problem like this, I

00:00:14.600 --> 00:00:20.320
like to expand out what 4,356
really is, so let me rewrite

00:00:20.320 --> 00:00:20.880
the number.

00:00:20.880 --> 00:00:21.940
So if I were to write it--
and I'll write it

00:00:21.940 --> 00:00:23.210
in different colors.

00:00:23.210 --> 00:00:33.420
So 4,356 is equal to--
and just think about

00:00:33.420 --> 00:00:34.360
how I just said it.

00:00:34.360 --> 00:00:48.190
It is equal to 4,000 plus
300 plus 50 plus 6.

00:00:48.190 --> 00:00:50.600
And you could come up with that
just based on how we said

00:00:50.600 --> 00:00:54.810
it: four thousand, three
hundred, and fifty-six.

00:00:54.810 --> 00:00:58.060
Now another way to think about
this is this is just like

00:00:58.060 --> 00:01:07.520
saying this is 4 thousands
plus-- or you could even think

00:01:07.520 --> 00:01:16.880
of "and"-- so plus 3 hundreds
plus 50, you could think of it

00:01:16.880 --> 00:01:22.040
as 5 tens plus 6.

00:01:22.040 --> 00:01:24.420
And instead of 6, we could
say plus 6 ones.

00:01:27.570 --> 00:01:32.620
And so if we go back to the
original number 4,356, this is

00:01:32.620 --> 00:01:36.440
the same thing as 4--
I'll write it down.

00:01:36.440 --> 00:01:39.230
Let me see how well I can-- I'll
write it up like this.

00:01:39.230 --> 00:01:51.020
This is the same thing is 4
thousands, 3 hundreds, 5 tens

00:01:51.020 --> 00:01:53.670
and then 6 ones.

00:01:53.670 --> 00:01:59.820
So when they ask what is the
place value of 3 into 4,356,

00:01:59.820 --> 00:02:02.120
we're concerned with
this 3 right here,

00:02:02.120 --> 00:02:03.260
and it's place value.

00:02:03.260 --> 00:02:05.090
It's in the hundreds place.

00:02:05.090 --> 00:02:06.970
If there was a 4 here, that
would mean we're dealing with

00:02:06.970 --> 00:02:07.850
4 hundreds.

00:02:07.850 --> 00:02:09.850
If there's a 5, 5 hundreds.

00:02:09.850 --> 00:02:12.260
It's the third from the right.

00:02:12.260 --> 00:02:13.410
This is the ones place.

00:02:13.410 --> 00:02:16.500
That's 6 ones, 5 tens,
3 hundreds.

00:02:16.500 --> 00:02:20.190
So the answer here is it is
in the hundreds place.


Writing a number in standard form

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:05.020
Write six hundred forty-five
million five hundred

00:00:05.020 --> 00:00:08.150
eighty-four thousand
four hundred

00:00:08.150 --> 00:00:10.600
sixty-two in standard form.

00:00:10.600 --> 00:00:13.000
So let's tackle this
piece by piece.

00:00:13.000 --> 00:00:16.059
So the first part we
have six hundred

00:00:16.059 --> 00:00:18.703
and forty-five million.

00:00:21.660 --> 00:00:22.580
So let's think about that.

00:00:22.580 --> 00:00:23.900
So we have six hundred
and forty-five.

00:00:26.940 --> 00:00:28.790
But it's not just six
hundred forty-five.

00:00:28.790 --> 00:00:31.540
We have six hundred forty-five
millions.

00:00:31.540 --> 00:00:37.960
So we could view that as
645 times 1,000,000.

00:00:37.960 --> 00:00:41.410
One million is one followed
by six zeroes.

00:00:41.410 --> 00:00:44.940
So this piece right here is
this right over here.

00:00:44.940 --> 00:00:48.840
That is six hundred and
forty-five million.

00:00:48.840 --> 00:00:50.350
And what is that when
we write it out?

00:00:50.350 --> 00:00:53.675
If we were to multiply this
out, this is equal to-- it

00:00:53.675 --> 00:00:57.800
would be 645 times this 1 with
six zeroes behind it.

00:00:57.800 --> 00:01:01.920
So this would be equal to 600--
I'll write it like this.

00:01:01.920 --> 00:01:07.690
This is equal to 645, and then
we have our six zeroes.

00:01:07.690 --> 00:01:10.560
One, two, three, one,
two, three.

00:01:10.560 --> 00:01:13.860
That's just this part
of the number.

00:01:13.860 --> 00:01:16.510
And I'm going to do it kind
of slow and do all of the

00:01:16.510 --> 00:01:18.710
different parts of the problem,
but once you get some

00:01:18.710 --> 00:01:20.610
practice, you'll find that these
are a bit second nature,

00:01:20.610 --> 00:01:22.240
and you won't have to go through
all of these steps.

00:01:22.240 --> 00:01:24.240
You'll just be able to
write the number.

00:01:24.240 --> 00:01:25.790
Now, let's move on
to the next part.

00:01:25.790 --> 00:01:28.680
We have five hundred
and eighty-four.

00:01:28.680 --> 00:01:30.010
Let me write that down.

00:01:33.700 --> 00:01:34.300
Well, I'll write it like this.

00:01:34.300 --> 00:01:40.070
584 thousand.

00:01:40.070 --> 00:01:42.100
So let me write the thousand.

00:01:42.100 --> 00:01:43.760
So 584 thousand.

00:01:43.760 --> 00:01:47.300
So it's 584 times 1,000.

00:01:47.300 --> 00:01:48.870
And what's that going
to look like?

00:01:48.870 --> 00:01:51.600
So that's this whole
part right here.

00:01:51.600 --> 00:01:55.400
So 584 times 1,000
is equal to what?

00:01:55.400 --> 00:01:59.840
Well, it's going to be 584 with
three zeroes behind it,

00:01:59.840 --> 00:02:01.863
or you could view it as 584
times the 1, and then you're

00:02:01.863 --> 00:02:04.010
going to have three zeroes
in the final answer.

00:02:04.010 --> 00:02:11.620
So it's going to be 584,000.

00:02:11.620 --> 00:02:13.400
We have our three zeroes
at the end.

00:02:13.400 --> 00:02:15.030
584, three zeroes.

00:02:15.030 --> 00:02:16.250
So that's that part.

00:02:16.250 --> 00:02:22.910
And then finally, we have four
hundred sixty-two, and that's

00:02:22.910 --> 00:02:24.960
just 462, straight up.

00:02:24.960 --> 00:02:28.330
You could view it as 462 ones.

00:02:28.330 --> 00:02:31.280
So then you just
have 462, which

00:02:31.280 --> 00:02:35.280
obviously equals just 462.

00:02:35.280 --> 00:02:38.430
Now our number is all
of these combined.

00:02:38.430 --> 00:02:45.450
It is 645,000,000 and
584,000 and 462.

00:02:45.450 --> 00:02:48.000
So one way to think about it
is that you could add these

00:02:48.000 --> 00:02:49.330
three numbers.

00:02:49.330 --> 00:03:08.300
So if we were to add them,
we get 645,584,462.

00:03:08.300 --> 00:03:10.570
Now, I said it'll become a
little bit of second nature to

00:03:10.570 --> 00:03:13.230
you in a little bit, and the way
to think about that, the

00:03:13.230 --> 00:03:16.040
easiest way to think about that,
is millions will have

00:03:16.040 --> 00:03:18.600
six zeroes behind them,
thousands will have three

00:03:18.600 --> 00:03:22.290
zeroes behind them, and just
regular numbers have no zeroes

00:03:22.290 --> 00:03:23.350
behind them.

00:03:23.350 --> 00:03:25.940
So what you can do when you
kind of want to learn it

00:03:25.940 --> 00:03:28.020
second nature, you'll just look
at the six hundred and

00:03:28.020 --> 00:03:33.060
forty-five million, so you'll
write 645, and you'll kind of

00:03:33.060 --> 00:03:35.320
keep in the back of your mind
that you're going to have to

00:03:35.320 --> 00:03:38.310
have six more digits to
the right of this.

00:03:38.310 --> 00:03:41.720
And then you say five hundred
and eighty-four thousand, so

00:03:41.720 --> 00:03:44.030
then you have your five hundred
and eighty-four

00:03:44.030 --> 00:03:50.320
thousand, so then you write that
down, 584,000, keeping in

00:03:50.320 --> 00:03:51.620
mind that you're going to
need three digits to

00:03:51.620 --> 00:03:52.530
the right of that.

00:03:52.530 --> 00:03:55.470
And so if they didn't tell us
462, we would just put three

00:03:55.470 --> 00:03:58.270
zeroes here, and that would also
fulfill the six zeroes we

00:03:58.270 --> 00:04:00.490
need behind 645,000,000.

00:04:00.490 --> 00:04:04.830
But then they tell us 462, so we
just write that right here.

00:04:04.830 --> 00:04:06.350
462.


Rounding whole numbers example 1

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:08.150
Round 24,259 to the
nearest hundred.

00:00:08.150 --> 00:00:09.590
You're going to find that
doing these problems are

00:00:09.590 --> 00:00:11.910
pretty straightforward, but
what I want to do is just

00:00:11.910 --> 00:00:15.750
think about what it means to
round to the nearest hundred.

00:00:15.750 --> 00:00:18.140
So what I'm going to do is I'm
going to draw a number line.

00:00:18.140 --> 00:00:21.220
Let me draw a number line here,
and I'm just going to

00:00:21.220 --> 00:00:23.970
mark off the hundreds
on the number line.

00:00:23.970 --> 00:00:37.290
So maybe we have 24,100, and
then we go to 24,200, then we

00:00:37.290 --> 00:00:44.490
go to 24,300, and then
we go to 24,400.

00:00:44.490 --> 00:00:46.000
I think you see what I
mean when I'm only

00:00:46.000 --> 00:00:47.250
marking off the hundreds.

00:00:47.250 --> 00:00:50.160
I'm going up by increments
of 100.

00:00:50.160 --> 00:00:56.130
Now, on this number line,
where is 24,259?

00:00:56.130 --> 00:01:00.200
So if we look at the number
line, it's more than 24,200

00:01:00.200 --> 00:01:02.930
and it's less than 24,300.

00:01:02.930 --> 00:01:09.220
And it's 259, so if this
distance right here is 100, 59

00:01:09.220 --> 00:01:14.300
is right about there, so that
is where our number is.

00:01:14.300 --> 00:01:18.570
That is 24,259.

00:01:18.570 --> 00:01:22.110
So when someone asks you to
round to the nearest hundred,

00:01:22.110 --> 00:01:25.700
they're literally saying round
to one of these increments of

00:01:25.700 --> 00:01:29.370
100 or round to whichever
increment of 100 that it is

00:01:29.370 --> 00:01:30.900
closest to.

00:01:30.900 --> 00:01:33.420
And if you look at it right like
this, if you just eyeball

00:01:33.420 --> 00:01:38.760
it, you'll actually see that it
is closer to 24,300 than it

00:01:38.760 --> 00:01:41.220
is to 24,200.

00:01:41.220 --> 00:01:45.160
So when you round it,
you round to 24,300.

00:01:45.160 --> 00:01:47.890
So if you round to the nearest
hundred, the answer

00:01:47.890 --> 00:01:51.180
literally is 24,300.

00:01:51.180 --> 00:01:53.310
Now that's kind of the
conceptual understanding of

00:01:53.310 --> 00:01:55.230
why it's even called the
nearest hundred.

00:01:55.230 --> 00:01:58.200
The nearest hundred is 24,300.

00:01:58.200 --> 00:01:59.870
But every time you do a problem
like this, you don't

00:01:59.870 --> 00:02:02.460
have to draw a number line and
go through this whole process,

00:02:02.460 --> 00:02:04.580
although you might want
to think about it.

00:02:04.580 --> 00:02:07.910
An easier process, or maybe a
more mechanical process, is

00:02:07.910 --> 00:02:14.900
you literally look at
the number 24,259.

00:02:14.900 --> 00:02:17.470
We want to round to the nearest
hundred, so you look

00:02:17.470 --> 00:02:19.080
at the hundreds place.

00:02:19.080 --> 00:02:21.860
This is the hundreds place right
here, and when we round,

00:02:21.860 --> 00:02:23.130
that means we don't
want any digits.

00:02:23.130 --> 00:02:25.650
We only want zeroes after
the hundreds place.

00:02:25.650 --> 00:02:28.490
So what you do is you look at
the place one less than the

00:02:28.490 --> 00:02:29.510
place you're rounding to.

00:02:29.510 --> 00:02:33.120
This is the hundreds place so
you look at the 5 right there,

00:02:33.120 --> 00:02:37.950
and if this number is 5 or
greater, if it's 5, 6, 7, 8,

00:02:37.950 --> 00:02:40.360
or 9, you round up.

00:02:40.360 --> 00:02:46.290
So 5 or greater, you round up.

00:02:49.910 --> 00:02:52.800
And so rounding up in this
situation, it is 5.

00:02:52.800 --> 00:02:56.640
It is 5 or greater, so rounding
up means that we go

00:02:56.640 --> 00:03:04.270
to 24,000, and since we're
rounding up, we make

00:03:04.270 --> 00:03:05.820
the 2 into a 3.

00:03:05.820 --> 00:03:13.540
We increment it by one, so
rounding up, so 24,300.

00:03:13.540 --> 00:03:15.570
That's what we mean
by rounding up.

00:03:15.570 --> 00:03:18.180
And just as kind of a
counterexample, if I had

00:03:18.180 --> 00:03:28.610
24,249 and I wanted to round
to the nearest hundred, I

00:03:28.610 --> 00:03:30.580
would say, OK, I want to round
to the nearest hundred.

00:03:30.580 --> 00:03:33.430
Let me look at the tens
place, this place one

00:03:33.430 --> 00:03:35.150
level to the right.

00:03:35.150 --> 00:03:38.745
It is not 5 or greater,
so I will round down.

00:03:43.010 --> 00:03:44.640
And when you round
down, be careful.

00:03:44.640 --> 00:03:46.120
It doesn't mean you
decreases this 2.

00:03:46.120 --> 00:03:48.700
It literally means you
just only have the 2.

00:03:48.700 --> 00:03:50.410
Just get rid of everything
after it.

00:03:50.410 --> 00:03:53.820
So it becomes 24,200.

00:03:53.820 --> 00:03:55.580
That's the process where
you round down.

00:03:55.580 --> 00:03:58.300
If you round up, it
becomes 24,300.

00:03:58.300 --> 00:03:59.180
And it makes sense.

00:03:59.180 --> 00:04:05.410
24,249 is going to be sitting
right over here someplace, so

00:04:05.410 --> 00:04:08.730
it's going to be closer
to 24,200.

00:04:08.730 --> 00:04:12.310
24,200 would be the nearest
hundred when we round

00:04:12.310 --> 00:04:13.470
down in this case.

00:04:13.470 --> 00:04:17.390
For the case of the problem,
24,259, the

00:04:17.390 --> 00:04:20.140
nearest hundred is 24,300.

00:04:20.140 --> 00:04:21.970
We round up.


Writing a number in expanded form

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:06.050
Write 14,897 in expanded form.

00:00:06.050 --> 00:00:08.165
Let me just rewrite the number,
and I'll color code

00:00:08.165 --> 00:00:10.650
it, and that way, we can keep
track of our digits.

00:00:10.650 --> 00:00:14.470
So we have 14,000.

00:00:14.470 --> 00:00:15.430
I don't have to write
it-- well, let me

00:00:15.430 --> 00:00:16.410
write it that big.

00:00:16.410 --> 00:00:31.150
14,000, 800, and 97-- I already
used the blue; maybe I

00:00:31.150 --> 00:00:35.990
should use yellow--
in expanded form.

00:00:35.990 --> 00:00:38.070
So let's think about
what place each of

00:00:38.070 --> 00:00:39.480
these digits are in.

00:00:39.480 --> 00:00:42.160
This right here, the 7,
is in the ones place.

00:00:47.100 --> 00:00:49.450
The 9 is in the tens place.

00:00:52.320 --> 00:00:55.556
This literally represents 9
tens, and we're going to see

00:00:55.556 --> 00:00:56.080
this in a second.

00:00:56.080 --> 00:00:59.250
This literally represents
7 ones.

00:00:59.250 --> 00:01:01.500
The 8 is in the hundreds
place.

00:01:04.310 --> 00:01:06.500
The 4 is in the thousands
place.

00:01:06.500 --> 00:01:12.170
It literally represents 4,000.

00:01:12.170 --> 00:01:14.800
And then the 1 is in the
ten-thousands place.

00:01:18.430 --> 00:01:21.210
And you see, every time you move
to the left, you move one

00:01:21.210 --> 00:01:23.540
place to the left, you're
multiplying by 10.

00:01:23.540 --> 00:01:26.260
Ones place, tens place, hundreds
place, thousands

00:01:26.260 --> 00:01:28.500
place, ten-thousands place.

00:01:28.500 --> 00:01:30.570
Now let's think about what
that really means.

00:01:30.570 --> 00:01:33.940
If this 1 is in the
ten-thousands place, that

00:01:33.940 --> 00:01:40.790
means that it literally
represents-- I want to do this

00:01:40.790 --> 00:01:42.520
in a way that my arrows
don't get mixed up.

00:01:42.520 --> 00:01:44.020
Actually, let me start
at the other end.

00:01:44.020 --> 00:01:45.970
Let me start with what
the 7 represents.

00:01:45.970 --> 00:01:49.820
The 7 literally represents
7 ones.

00:01:52.350 --> 00:01:55.510
Or another way to think about
it, you could say it

00:01:55.510 --> 00:01:58.920
represents 7 times 1.

00:01:58.920 --> 00:02:00.000
All of these are equivalent.

00:02:00.000 --> 00:02:01.150
They represent 7 ones.

00:02:01.150 --> 00:02:02.640
Now let's think about the 9.

00:02:02.640 --> 00:02:04.450
That's why I'm doing it from the
right, so that the arrows

00:02:04.450 --> 00:02:06.220
don't have to cross
each other.

00:02:06.220 --> 00:02:07.385
So what does the 9 represent?

00:02:07.385 --> 00:02:10.720
It represents 9 tens.

00:02:10.720 --> 00:02:13.530
You could literally imagine
you have 9 actual tens.

00:02:13.530 --> 00:02:16.270
You could have a 10, plus
a 10, plus a 10.

00:02:16.270 --> 00:02:17.820
Do that nine times.

00:02:17.820 --> 00:02:22.300
That's literally what it
represents: 9 actual tens.

00:02:22.300 --> 00:02:27.720
9 tens, or you could say it's
the same thing as 9 times 10,

00:02:27.720 --> 00:02:30.280
or 90, either way you want
to think about it.

00:02:30.280 --> 00:02:32.060
So let me write all
the different ways

00:02:32.060 --> 00:02:33.620
to think about it.

00:02:33.620 --> 00:02:36.330
It represents all of these
things: 9 tens, or 9

00:02:36.330 --> 00:02:39.100
times 10, or 90.

00:02:39.100 --> 00:02:41.430
So then we have our 8.

00:02:41.430 --> 00:02:45.250
Our 8 represents-- we see it's
in the hundreds place.

00:02:45.250 --> 00:02:46.570
It represents 8 hundreds.

00:02:49.880 --> 00:02:53.210
Or you could view that as being
equivalent to 8 times

00:02:53.210 --> 00:02:56.570
100-- a hundred, not
a thousand-- 8

00:02:56.570 --> 00:02:59.140
times 100, or 800.

00:02:59.140 --> 00:03:03.130
That 8 literally represents
8 hundreds, 800.

00:03:03.130 --> 00:03:04.540
And then the 4.

00:03:04.540 --> 00:03:06.350
I think you get the idea here.

00:03:06.350 --> 00:03:08.110
This represents the
thousands place.

00:03:08.110 --> 00:03:14.270
It represents 4 thousands, which
is the same thing as 4

00:03:14.270 --> 00:03:18.950
times 1,000, which is the
same thing as 4,000.

00:03:18.950 --> 00:03:22.350
4,000 is the same thing
as 4 thousands.

00:03:22.350 --> 00:03:23.420
Add it up.

00:03:23.420 --> 00:03:26.520
And then finally, we have this
1, which is sitting in the

00:03:26.520 --> 00:03:31.730
ten-thousands place, so it
literally represents 1

00:03:31.730 --> 00:03:32.980
ten-thousand.

00:03:39.270 --> 00:03:42.010
You can imagine if these were
chips, kind of poker chips,

00:03:42.010 --> 00:03:44.700
that would represent one of the
blue poker chips and each

00:03:44.700 --> 00:03:46.700
blue poker chip represents
10,000.

00:03:46.700 --> 00:03:48.000
I don't know if that
helps you or not.

00:03:48.000 --> 00:03:52.330
And 1 ten-thousand is the same
thing as 1 times 10,000 which

00:03:52.330 --> 00:03:55.020
is the same thing as 10,000.

00:03:55.020 --> 00:03:58.020
So when they ask us to write it
in expanded form, we could

00:03:58.020 --> 00:04:03.020
write 14,897 literally as the
sum of these numbers, of its

00:04:03.020 --> 00:04:04.270
components, or we could
write it as the

00:04:04.270 --> 00:04:05.820
sum of these numbers.

00:04:05.820 --> 00:04:06.740
Actually, let me write this.

00:04:06.740 --> 00:04:09.340
This top 7 times 1 is
just equal to 7.

00:04:09.340 --> 00:04:29.710
So 14,897 is the same thing as
10,000 plus 4,000 plus 800

00:04:29.710 --> 00:04:39.220
plus 90 plus 7.

00:04:39.220 --> 00:04:41.620
So you could consider this
expanded form, or you could

00:04:41.620 --> 00:04:44.330
use this version of it, or you
could say this the same thing

00:04:44.330 --> 00:04:48.860
as 1 times 10,000, depending on
what people consider to be

00:04:48.860 --> 00:05:02.860
expanded form-- plus 4 times
1,000 plus 8 times 100 plus 9

00:05:02.860 --> 00:05:09.160
times 10 plus 7 times 1.

00:05:09.160 --> 00:05:10.930
I'll scroll to the right
a little bit.

00:05:14.500 --> 00:05:18.910
So either of these could be
considered expanded form.


Rounding whole numbers 3

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.990
Round 1,585 to the
nearest ten.

00:00:04.990 --> 00:00:06.290
Let me rewrite the number.

00:00:06.290 --> 00:00:11.500
1,585 to the nearest ten.

00:00:11.500 --> 00:00:13.860
So we want to focus
on the tens place.

00:00:16.610 --> 00:00:24.240
If we were to round it up, we
would go to 1,500-- let me

00:00:24.240 --> 00:00:24.950
color code it.

00:00:24.950 --> 00:00:29.162
We would round up to 1,590.

00:00:29.162 --> 00:00:31.590
The 8 would round up to a 9.

00:00:31.590 --> 00:00:38.425
And 90, if we were to round
down, we would go to 1,500,

00:00:38.425 --> 00:00:41.190
and we would round down to 80.

00:00:41.190 --> 00:00:43.560
So these are our two choices.

00:00:43.560 --> 00:00:45.370
Now we've seen the last
two examples.

00:00:45.370 --> 00:00:47.590
If the place to the right
of it is 5 or

00:00:47.590 --> 00:00:50.790
greater, we round up.

00:00:50.790 --> 00:00:52.340
Well, the place to the
right of it is

00:00:52.340 --> 00:00:54.010
definitely 5 or greater.

00:00:54.010 --> 00:00:56.130
It is 5, so we round up.

00:00:56.130 --> 00:01:00.890
So it is 1,590.

00:01:00.890 --> 00:01:02.420
And we're done!


Rounding whole numbers example 2

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:08.500
Round 423,275 to the
nearest thousand.

00:00:08.500 --> 00:00:15.360
So let me rewrite it: 423,275.

00:00:15.360 --> 00:00:20.080
And so the thousands place is
the 3 right here, and so if we

00:00:20.080 --> 00:00:24.920
were round it up to the nearest
thousand, we would go

00:00:24.920 --> 00:00:29.990
to 420-- let me write it so we
just focus on the 3-- we would

00:00:29.990 --> 00:00:39.160
go up to 424,000 if we wanted
to round up, 424,000, and if

00:00:39.160 --> 00:00:46.950
we wanted to round down,
we would go to 423,000.

00:00:46.950 --> 00:00:49.990
We would get rid of the 275.

00:00:49.990 --> 00:00:52.010
423,000.

00:00:52.010 --> 00:00:53.510
So this is our choice.

00:00:53.510 --> 00:00:58.140
Round up to 424,000 or round
down to 423,000.

00:00:58.140 --> 00:01:04.560
And to figure it out, we just
look at the digit one place to

00:01:04.560 --> 00:01:08.070
the right of the 3, so we look
at the 2 right there.

00:01:08.070 --> 00:01:11.350
If that digit is 5 or greater,
you round up.

00:01:11.350 --> 00:01:13.100
So this is 5.

00:01:13.100 --> 00:01:17.220
So if this is greater than
or equal to 5, 5 or

00:01:17.220 --> 00:01:18.880
greater, you round up.

00:01:18.880 --> 00:01:22.550
If it's less than 5,
you round down.

00:01:22.550 --> 00:01:28.460
2 is definitely less than 5, so
we just round down, so it

00:01:28.460 --> 00:01:31.600
is 423,000.

00:01:31.600 --> 00:01:34.260
Now just to visualize what
this means to the nearest

00:01:34.260 --> 00:01:36.380
thousand, if I were to do a
number line-- and you don't

00:01:36.380 --> 00:01:36.930
have to do this.

00:01:36.930 --> 00:01:39.470
We've gotten the answer, but
just to have a little bit

00:01:39.470 --> 00:01:42.390
better visualization of it,
if I were to increment by

00:01:42.390 --> 00:01:52.360
thousands, you might have
422,000, 423,000.

00:01:52.360 --> 00:01:58.940
You have 424,000, and then
maybe over here, you have

00:01:58.940 --> 00:02:02.690
425,000, and you could
keep going.

00:02:02.690 --> 00:02:10.820
Now 423,275 is going to be
someplace right around here.

00:02:10.820 --> 00:02:13.730
And so when we round to the
nearest thousand, we have to

00:02:13.730 --> 00:02:15.120
pick between that and that.

00:02:15.120 --> 00:02:19.480
We see that it much closer to
423,000 than to 424,000, so we

00:02:19.480 --> 00:02:20.500
round it right there.

00:02:20.500 --> 00:02:23.220
But you just use the rules we
just came up with, and we

00:02:23.220 --> 00:02:26.220
rounded down to 423,000.


Optimal angle for a projectile part 1: Components of initial velocity

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.770
Let's say we're going to
shoot some object into

00:00:02.770 --> 00:00:04.220
the air at an angle.

00:00:04.220 --> 00:00:09.075
Let's say its speed is s and
the angle at which we shoot

00:00:09.075 --> 00:00:14.090
it, the angle above the
horizontal is theta.

00:00:14.090 --> 00:00:16.780
What I want to do in this video
is figure out how far

00:00:16.780 --> 00:00:20.990
this object is going to travel
as a function of the angle and

00:00:20.990 --> 00:00:22.770
as a function of the speed, but
we're going to assume that

00:00:22.770 --> 00:00:23.590
we're given the speed.

00:00:23.590 --> 00:00:25.250
That that's a bit
of a constant.

00:00:25.250 --> 00:00:30.120
So if this is the ground right
here, we want to figure out

00:00:30.120 --> 00:00:31.920
how far this thing is
going to travel.

00:00:31.920 --> 00:00:34.800
So you can imagine, it's going
to travel in this parabolic

00:00:34.800 --> 00:00:40.050
path and land at some
point out there.

00:00:40.050 --> 00:00:44.000
And so if this is at distance 0,
we could call this distance

00:00:44.000 --> 00:00:46.950
out here distance d.

00:00:46.950 --> 00:00:50.090
Now whenever you do any problem
like this where you're

00:00:50.090 --> 00:00:53.750
shooting something off at an
angle, the best first step is

00:00:53.750 --> 00:00:55.640
to break down that vector.

00:00:55.640 --> 00:00:57.235
Remember, a vector is something
that has magnitude

00:00:57.235 --> 00:00:58.210
and direction.

00:00:58.210 --> 00:01:00.240
The magnitude is s.

00:01:00.240 --> 00:01:02.960
Maybe feet per second
or miles per hour.

00:01:02.960 --> 00:01:04.459
And the direction is theta.

00:01:04.459 --> 00:01:07.820
So if you have s and theta,
you're giving me a vector.

00:01:07.820 --> 00:01:09.900
And so what you want to do is
you want to break this vector

00:01:09.900 --> 00:01:13.370
down into its vertical and
horizontal components first

00:01:13.370 --> 00:01:14.700
and then deal with
them separately.

00:01:14.700 --> 00:01:16.560
One, to help you figure out how
long you're in the air.

00:01:16.560 --> 00:01:18.420
And then, the other to
figure out how far

00:01:18.420 --> 00:01:20.040
you actually travel.

00:01:20.040 --> 00:01:26.050
So let me make a big version
of the vector right there.

00:01:26.050 --> 00:01:28.590
Once again, the magnitude
of the vector is s.

00:01:28.590 --> 00:01:31.970
So you could imagine that the
length of this arrow is s.

00:01:31.970 --> 00:01:34.800
And this angle right
here is theta.

00:01:37.390 --> 00:01:39.550
And to break it down into its
horizontal and vertical

00:01:39.550 --> 00:01:42.360
components, we just set up a
right triangle and just use

00:01:42.360 --> 00:01:44.770
our basic trig ratios.

00:01:44.770 --> 00:01:46.030
So let me do that.

00:01:46.030 --> 00:01:48.490
So this is the ground
right there.

00:01:48.490 --> 00:01:53.850
I can drop a vertical from the
tip of that arrow to set up a

00:01:53.850 --> 00:01:55.100
right triangle.

00:01:55.100 --> 00:01:59.560
And the length of the-- or the
magnitude of the vertical

00:01:59.560 --> 00:02:02.910
component of our velocity
is going to be this

00:02:02.910 --> 00:02:04.160
length right here.

00:02:06.810 --> 00:02:08.930
That is going to be-- you could
imagine, the length of

00:02:08.930 --> 00:02:11.790
that is going to be our
vertical speed.

00:02:11.790 --> 00:02:14.590
So this is our vertical speed.

00:02:14.590 --> 00:02:19.880
Maybe I'll just call that
the speed sub vertical.

00:02:19.880 --> 00:02:24.860
And then, this right here, the
length of this part of the

00:02:24.860 --> 00:02:27.160
triangle-- let me do that
in a different color.

00:02:27.160 --> 00:02:31.880
The length of this part of the
triangle is going to be our

00:02:31.880 --> 00:02:35.890
horizontal speed, or the
component of this velocity in

00:02:35.890 --> 00:02:37.770
the horizontal direction.

00:02:37.770 --> 00:02:40.750
And I use this word velocity
when I specify

00:02:40.750 --> 00:02:42.940
a speed and a direction.

00:02:42.940 --> 00:02:45.310
Speed is just the magnitude
of the velocity.

00:02:45.310 --> 00:02:49.380
So the magnitude of this side
is going to be speed

00:02:49.380 --> 00:02:50.470
horizontal.

00:02:50.470 --> 00:02:52.370
And to figure it out,
you literally use

00:02:52.370 --> 00:02:54.280
our basic trig ratios.

00:02:54.280 --> 00:02:55.490
So we have a right triangle.

00:02:55.490 --> 00:02:56.800
This is the hypotenuse.

00:02:56.800 --> 00:02:59.210
And we could write down
soh cah toa up here.

00:03:01.800 --> 00:03:04.000
Let me write it down
in yellow.

00:03:04.000 --> 00:03:08.880
soh cah toa.

00:03:08.880 --> 00:03:11.960
And this tells us that sine is
opposite over hypotenuse,

00:03:11.960 --> 00:03:15.680
cosine is adjacent over
hypotenuse and tangent is

00:03:15.680 --> 00:03:16.690
opposite over adjacent.

00:03:16.690 --> 00:03:17.690
So let's see what we can do.

00:03:17.690 --> 00:03:20.020
We're assuming we know
theta, we know s.

00:03:20.020 --> 00:03:22.480
We want to figure out what the
vertical and the horizontal

00:03:22.480 --> 00:03:23.690
components are.

00:03:23.690 --> 00:03:28.110
So what's the vertical component
going to be?

00:03:28.110 --> 00:03:34.150
Well the vertical component
is opposite this theta.

00:03:34.150 --> 00:03:38.320
But we know the hypotenuse is
s, so we could use sine

00:03:38.320 --> 00:03:40.790
because that deals with the
opposite and the hypotenuse.

00:03:40.790 --> 00:03:45.610
And the sine function tells us
that sine of theta-- actually,

00:03:45.610 --> 00:03:47.710
let me do this in green since
we're doing all the vertical

00:03:47.710 --> 00:03:49.060
stuff in green.

00:03:49.060 --> 00:03:55.560
Sine of theta is going to be
equal to opposite, which is

00:03:55.560 --> 00:03:58.620
the magnitude of our
vertical velocity.

00:03:58.620 --> 00:04:02.520
So the opposite side is this
side right here, over our

00:04:02.520 --> 00:04:03.270
hypotenuse.

00:04:03.270 --> 00:04:06.430
And our hypotenuse
is the speed s.

00:04:06.430 --> 00:04:10.150
And so if we want to solve for
our vertical velocity or the

00:04:10.150 --> 00:04:13.310
vertical component of our
velocity, we multiply both

00:04:13.310 --> 00:04:15.250
sides of this equation by s.

00:04:15.250 --> 00:04:20.990
So you get s sine of theta
is equal to the vertical

00:04:20.990 --> 00:04:27.010
component of our velocity,
s sine of theta.

00:04:27.010 --> 00:04:29.450
And now for the horizontal
component we do the same

00:04:29.450 --> 00:04:31.830
thing, but we don't
use sine anymore.

00:04:31.830 --> 00:04:34.430
This is now adjacent
to the angle.

00:04:34.430 --> 00:04:37.290
So cosine deals with the
adjacent side and the

00:04:37.290 --> 00:04:38.140
hypotenuse.

00:04:38.140 --> 00:04:42.580
So we could say that the cosine
of theta is equal to

00:04:42.580 --> 00:04:48.790
the adjacent side to the angle,
that is the horizontal

00:04:48.790 --> 00:04:52.560
speed, over the hypotenuse.

00:04:52.560 --> 00:04:55.520
The hypotenuse is this length
right here, over s.

00:04:55.520 --> 00:04:59.440
So if we want to solve for the
horizontal speed or the

00:04:59.440 --> 00:05:02.550
horizontal component or the
magnitude of the horizontal

00:05:02.550 --> 00:05:05.720
component, we'd just multiply
both sides times s.

00:05:05.720 --> 00:05:11.980
And you get s cosine of
theta is equal to

00:05:11.980 --> 00:05:13.970
the horizontal component.

00:05:13.970 --> 00:05:18.860
So we now know how fast we are
travelling in this direction,

00:05:18.860 --> 00:05:20.860
in the horizontal component.

00:05:20.860 --> 00:05:25.140
We know that that is going
to be s cosine of theta.

00:05:25.140 --> 00:05:31.200
And we know in the vertical
direction-- let me do that in

00:05:31.200 --> 00:05:38.470
the vertical direction, the
magnitude is s sine of theta.

00:05:38.470 --> 00:05:41.560
It is s sine of theta.

00:05:41.560 --> 00:05:44.120
So now that we've broken up into
the two components, we're

00:05:44.120 --> 00:05:48.880
ready to figure out how long
we're going to be in the air.


Optimal angle for a projectile part 2: Hangtime

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.420
Let's figure out how long this
object is going to be in the

00:00:03.420 --> 00:00:07.590
air given that its vertical
velocity, or the magnitude of

00:00:07.590 --> 00:00:11.370
the vertical velocity
is s sine of theta.

00:00:11.370 --> 00:00:17.540
So its speed in the vertical
direction is s sine of theta.

00:00:17.540 --> 00:00:19.440
So how long is it going
to be in the air?

00:00:19.440 --> 00:00:22.620
Well if I told you that
something is going upwards at

00:00:22.620 --> 00:00:27.030
10 meters per second and gravity
is decelerating it at

00:00:27.030 --> 00:00:29.160
10 meters per second squared.

00:00:29.160 --> 00:00:31.420
So every second it's going to
slow it down by 10 meters per

00:00:31.420 --> 00:00:35.240
second, how long will it take
for that object to get to 0,

00:00:35.240 --> 00:00:36.020
to stop moving?

00:00:36.020 --> 00:00:36.810
Let me write that down.

00:00:36.810 --> 00:00:41.460
Let's say that some object is
moving upwards at 10 meters

00:00:41.460 --> 00:00:42.780
per second.

00:00:42.780 --> 00:00:47.196
And let's say that the gravity
is slowing it down.

00:00:52.500 --> 00:01:01.320
Slowing it down at 10
meters per second.

00:01:01.320 --> 00:01:04.540
So every second that goes by,
it'll slow this thing down by

00:01:04.540 --> 00:01:06.130
10 meters per second.

00:01:06.130 --> 00:01:09.820
Well, it'll take it exactly 1
second to make it go from 10

00:01:09.820 --> 00:01:12.620
meters per second to 0
meters per second.

00:01:12.620 --> 00:01:14.840
And then it's going to be at
some height in the air, and

00:01:14.840 --> 00:01:16.320
then the thing's going to
start accelerating.

00:01:16.320 --> 00:01:18.640
Gravity is going to start
accelerating it downward.

00:01:18.640 --> 00:01:21.690
And then it'll take another
second for it to go from 0--

00:01:21.690 --> 00:01:24.170
from having no velocity,
to having 10

00:01:24.170 --> 00:01:25.610
meters per second again.

00:01:25.610 --> 00:01:29.760
So in this case, it'll take the
time in the air-- we could

00:01:29.760 --> 00:01:34.960
say time sub air, I guess, is
going to be equal to this 10

00:01:34.960 --> 00:01:37.650
meters per second,
your velocity.

00:01:37.650 --> 00:01:43.530
10 meters per second divided
by the acceleration.

00:01:43.530 --> 00:01:46.250
Divided by this 10 meters
per second.

00:01:46.250 --> 00:01:50.140
10 meters per second times 2.

00:01:50.140 --> 00:01:52.880
This is how long it'll take for
the object to go from 10

00:01:52.880 --> 00:01:55.610
meters per second to 0 at
some point in the air.

00:01:55.610 --> 00:01:58.050
And then it's going to take the
exact same amount of time

00:01:58.050 --> 00:01:59.910
for it to fall back
to the ground.

00:01:59.910 --> 00:02:03.340
So times 2.

00:02:03.340 --> 00:02:06.350
If the object was moving upwards
at 20 meters per

00:02:06.350 --> 00:02:10.080
second and gravity is still
slowing it down at 10 meters

00:02:10.080 --> 00:02:14.330
per second per second, then it's
going to take 2 seconds.

00:02:14.330 --> 00:02:17.610
If this was 20, then
this would be 20.

00:02:17.610 --> 00:02:20.950
And it'll take 2 seconds to slow
it down to 0 and then 2

00:02:20.950 --> 00:02:23.670
more seconds until it hits
the ground again.

00:02:23.670 --> 00:02:26.560
For it to speed back up as
it approaches the ground.

00:02:26.560 --> 00:02:34.660
So no matter what your upward
velocity, the time in the air

00:02:34.660 --> 00:02:39.550
is going to be your speed, your
vertical speed, divided

00:02:39.550 --> 00:02:42.690
by the acceleration
of gravity.

00:02:42.690 --> 00:02:44.820
And this is the amount of time
it's going to take you to go

00:02:44.820 --> 00:02:46.660
from this point to that point.

00:02:46.660 --> 00:02:50.000
To have some vertical velocity
and then slow down to 0.

00:02:50.000 --> 00:02:51.960
And it's going to take the exact
same amount of time for

00:02:51.960 --> 00:02:54.400
you to speed back up by gravity
and get to your

00:02:54.400 --> 00:02:55.200
original speed.

00:02:55.200 --> 00:02:56.730
We're assuming no
air resistance.

00:02:56.730 --> 00:02:58.520
So it's kind of a
pure problem.

00:02:58.520 --> 00:03:00.860
So this is the time up, the time
down is going to be the

00:03:00.860 --> 00:03:01.350
same thing.

00:03:01.350 --> 00:03:03.900
So we can multiply that by 2.

00:03:03.900 --> 00:03:06.850
Now, we already know what the
vertical component for our

00:03:06.850 --> 00:03:07.710
problem is.

00:03:07.710 --> 00:03:09.960
It is s sine of theta.

00:03:09.960 --> 00:03:12.240
So we could just substitute
that back in there.

00:03:12.240 --> 00:03:15.260
And we know how long we're
going to be in the air.

00:03:15.260 --> 00:03:20.040
The time in the air is going to
be our speed-- or I should

00:03:20.040 --> 00:03:21.130
maybe put the 2 out front.

00:03:21.130 --> 00:03:23.665
2 times s sine theta.

00:03:26.810 --> 00:03:28.080
Let me make it clear.

00:03:28.080 --> 00:03:31.730
This 2 right here is
this 2 right there.

00:03:31.730 --> 00:03:37.550
All of that over the
acceleration of gravity.

00:03:37.550 --> 00:03:41.520
So if you told me that I'm
shooting this object off at--

00:03:41.520 --> 00:03:43.710
I don't know-- 100 meters
per second.

00:03:43.710 --> 00:03:48.660
So if this is 100 meters per
second and if theta were-- I

00:03:48.660 --> 00:03:54.390
don't know-- let's say theta
were 30 degrees, then sine of

00:03:54.390 --> 00:03:55.550
theta would be 1/2.

00:03:55.550 --> 00:03:59.120
So it'd be 100 meters per second
times 1/2 divided by

00:03:59.120 --> 00:04:01.960
the acceleration of gravity
times 2 would tell you exactly

00:04:01.960 --> 00:04:03.500
how long you would
be in the air.

00:04:03.500 --> 00:04:07.420
How long it takes to go up all
the way, become stationery,

00:04:07.420 --> 00:04:10.530
and then fall back down
to the ground.


Optimal angle for a projectile part 3: Horizontal distance as a function of angle (and speed)

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:03.750
We now know how long the object
is going to be in the

00:00:03.750 --> 00:00:05.500
air, so we're ready to
figure out how far

00:00:05.500 --> 00:00:06.860
it's going to travel.

00:00:06.860 --> 00:00:11.030
So we can just go back to kind
of the core formula in all of

00:00:11.030 --> 00:00:14.140
really kinematics, all of kind
of projectile motion or

00:00:14.140 --> 00:00:17.690
mechanical physical problems,
and that's distance is equal

00:00:17.690 --> 00:00:18.940
to rate times time.

00:00:23.460 --> 00:00:26.310
Now, we're talking about the
horizontal distance.

00:00:26.310 --> 00:00:29.170
So our distance is going to be
equal to-- what's our rate in

00:00:29.170 --> 00:00:33.060
the horizontal direction?

00:00:33.060 --> 00:00:36.960
We care about horizontal
distance traveled, so our rate

00:00:36.960 --> 00:00:41.550
needs to be the horizontal
component of the velocity, or

00:00:41.550 --> 00:00:43.190
the magnitude of the horizontal

00:00:43.190 --> 00:00:44.400
component of the velocity.

00:00:44.400 --> 00:00:46.430
And we figured that out
in the first video.

00:00:46.430 --> 00:00:49.540
That is s cosine of theta.

00:00:49.540 --> 00:00:51.440
So let's write that
down right here.

00:00:51.440 --> 00:00:59.610
So our rate is s cosine
of theta.

00:00:59.610 --> 00:01:01.850
And how long will we
be traveling at

00:01:01.850 --> 00:01:03.420
this horizontal speed?

00:01:05.930 --> 00:01:08.240
Well, we'll be going at
that speed as long as

00:01:08.240 --> 00:01:09.460
we are in the air.

00:01:09.460 --> 00:01:10.780
So how long are we in the air?

00:01:10.780 --> 00:01:12.630
Well, we figured that out
in the last video.

00:01:12.630 --> 00:01:16.600
We're going to be in the air
this long-- 2 s sine of theta

00:01:16.600 --> 00:01:18.210
divided by g.

00:01:18.210 --> 00:01:29.870
So the time is going to be
2 s sine of theta over g.

00:01:29.870 --> 00:01:32.170
So the total distance we're
going to travel, pretty

00:01:32.170 --> 00:01:34.320
straightforward, rate
times time.

00:01:34.320 --> 00:01:36.160
It's just the product
of these two things.

00:01:36.160 --> 00:01:39.230
And we could put all of the
constants out front, so it's a

00:01:39.230 --> 00:01:41.940
little bit clearer that it's
a function of theta.

00:01:41.940 --> 00:01:46.730
So we can write that the
distance traveled-- let me do

00:01:46.730 --> 00:01:48.360
that same green.

00:01:48.360 --> 00:01:52.670
The distance traveled as a
function of theta is equal

00:01:52.670 --> 00:01:55.790
to-- I'll do that
in this blue.

00:01:55.790 --> 00:02:00.460
This s times 2s divided by g
is-- I'll do it in a neutral

00:02:00.460 --> 00:02:01.020
color actually.

00:02:01.020 --> 00:02:08.220
This s times 2s divided by g is
2 times s squared over g.

00:02:08.220 --> 00:02:15.440
So 2s squared over g times
cosine of theta

00:02:15.440 --> 00:02:17.110
times sine of theta.

00:02:19.730 --> 00:02:21.450
So now we have a general
function.

00:02:21.450 --> 00:02:23.830
You give me an angle that I'm
going to shoot something off

00:02:23.830 --> 00:02:28.540
at and you give me the magnitude
of its velocity, and

00:02:28.540 --> 00:02:30.860
you give me the acceleration
of gravity.

00:02:30.860 --> 00:02:33.620
I guess if we were on some
other planet, who knows?

00:02:33.620 --> 00:02:35.350
And I will tell you
exactly what the

00:02:35.350 --> 00:02:38.180
horizontal distance is.


Optimal angle for a projectile part 4: Finding the optimal angle and distance with a bit of calculus

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.510
Now that we have distance
explicitly as a function of

00:00:04.510 --> 00:00:07.040
the angle that we're shooting
the object at, we can use a

00:00:07.040 --> 00:00:10.550
little bit of calculus to figure
out the optimal angle,

00:00:10.550 --> 00:00:13.290
the angle that's going optimize
our distance.

00:00:13.290 --> 00:00:17.340
And since we only care about
angles from 0 degrees to

00:00:17.340 --> 00:00:19.990
really 90 degrees, let's
constrain ourselves.

00:00:19.990 --> 00:00:21.870
So we're going to optimize
things for

00:00:21.870 --> 00:00:24.540
angles between 0 degrees.

00:00:24.540 --> 00:00:28.310
So theta is going to be greater
than or equal to 0 and

00:00:28.310 --> 00:00:31.490
less than or equal to 90.

00:00:31.490 --> 00:00:33.150
So let's see how we can do it.

00:00:33.150 --> 00:00:35.970
And just to get an idea of what
we're even conceptually

00:00:35.970 --> 00:00:38.000
doing with the calculus,
remember when you take a

00:00:38.000 --> 00:00:42.460
derivative, you are finding
the slope of a line, an

00:00:42.460 --> 00:00:44.760
instantaneous slope of a line.

00:00:44.760 --> 00:00:46.730
And if you were to graph this--
and I encourage you to

00:00:46.730 --> 00:00:49.960
graph it on your own, maybe with
a graphing calculator--

00:00:49.960 --> 00:00:53.850
it will look something like
this over the interval.

00:00:53.850 --> 00:00:58.560
It will look like this where
that is the distance as a

00:00:58.560 --> 00:01:02.410
function of theta axis
and then this would

00:01:02.410 --> 00:01:04.769
be our theta axis.

00:01:04.769 --> 00:01:07.810
And we care about angles between
0 and 90 degrees.

00:01:07.810 --> 00:01:11.670
So if you were to graph this
thing, so this is 0 degrees,

00:01:11.670 --> 00:01:14.620
this is maybe 90 degrees
right here.

00:01:14.620 --> 00:01:18.780
The graph of this function
will look like this.

00:01:18.780 --> 00:01:21.760
It'll look something
like this.

00:01:21.760 --> 00:01:24.510
It will look something
like that.

00:01:24.510 --> 00:01:27.900
And what we want to do is find
the angle, there's some angle

00:01:27.900 --> 00:01:32.720
here that gives us the
optimal distance.

00:01:32.720 --> 00:01:40.020
So this is, right here, this
is the optimal distance.

00:01:40.020 --> 00:01:41.560
And what we want to do
is find that out.

00:01:41.560 --> 00:01:43.570
And when you look at the graph,
and you could do it on

00:01:43.570 --> 00:01:46.640
a graphing calculator if you
like, what happens to the

00:01:46.640 --> 00:01:50.520
instantaneous slope at that
optimal distance?

00:01:50.520 --> 00:01:51.580
Well it's flat.

00:01:51.580 --> 00:01:53.130
The slope there is 0.

00:01:56.540 --> 00:01:59.580
So what we need to do is take
the derivative of this

00:01:59.580 --> 00:02:02.850
function and then just figure
out at what angle is the

00:02:02.850 --> 00:02:06.080
derivative or the instantaneous
slope of this

00:02:06.080 --> 00:02:08.539
function equal to 0?

00:02:08.539 --> 00:02:11.590
And then we're done We will know
this mystery angle, this

00:02:11.590 --> 00:02:14.220
optimal angle, to shoot
the object at.

00:02:14.220 --> 00:02:16.020
So let's take the derivative.

00:02:16.020 --> 00:02:19.860
So the derivative, we'll just
use our derivative rules here.

00:02:19.860 --> 00:02:23.840
The derivative of-- I will call
it d prime I guess, or we

00:02:23.840 --> 00:02:25.920
could say the derivative of the
distance with respect to

00:02:25.920 --> 00:02:29.470
theta is equal to-- we're
assuming that s and g are

00:02:29.470 --> 00:02:32.360
constants, so we don't have to
worry about them right now.

00:02:32.360 --> 00:02:35.220
We could just put them out front
since we're assuming

00:02:35.220 --> 00:02:36.170
they're constants.

00:02:36.170 --> 00:02:38.580
And then we can do the product
rule to take the derivative of

00:02:38.580 --> 00:02:42.700
this part with respect
to theta.

00:02:42.700 --> 00:02:45.510
In the product rule, we take
the derivative of the first

00:02:45.510 --> 00:02:48.150
function times the
second function.

00:02:48.150 --> 00:02:51.070
So the derivative of
cosine of theta is

00:02:51.070 --> 00:02:54.820
negative sine of theta.

00:02:54.820 --> 00:02:56.010
And we're going to
multiply that

00:02:56.010 --> 00:02:58.320
times the second function.

00:02:58.320 --> 00:03:01.290
So that's times the
sine of theta.

00:03:01.290 --> 00:03:06.880
And to that we're going to add
the first function, which is

00:03:06.880 --> 00:03:10.580
cosine of theta times
the derivative

00:03:10.580 --> 00:03:12.170
of the second function.

00:03:12.170 --> 00:03:17.250
The derivative of sine theta
is cosine of theta.

00:03:17.250 --> 00:03:18.610
I know it's a little
bit confusing.

00:03:18.610 --> 00:03:20.820
All we did is we took the
derivative of the first one

00:03:20.820 --> 00:03:22.190
times the second one.

00:03:22.190 --> 00:03:24.770
And then we took the derivative
of the second one

00:03:24.770 --> 00:03:26.440
times the first one.

00:03:26.440 --> 00:03:28.450
Let me make it even more
explicitly clear.

00:03:28.450 --> 00:03:31.290
We took the derivative of this
guy here, so this is the

00:03:31.290 --> 00:03:33.520
derivative with respect
to theta.

00:03:33.520 --> 00:03:37.530
And we took the derivative of
this guy over here with

00:03:37.530 --> 00:03:39.720
respect to theta.

00:03:39.720 --> 00:03:40.960
We took the derivative
of cosine there and

00:03:40.960 --> 00:03:42.190
multiplied it by sine.

00:03:42.190 --> 00:03:43.510
Took the derivative
of sine here and

00:03:43.510 --> 00:03:44.860
multiplied it by cosine.

00:03:44.860 --> 00:03:46.290
Just the product rule.

00:03:46.290 --> 00:03:47.130
Now what does this give us?

00:03:47.130 --> 00:03:49.500
We can simplify this
a good bit.

00:03:49.500 --> 00:03:54.440
So we could write the derivative
d prime is equal

00:03:54.440 --> 00:03:57.170
to-- we could keep this constant
out there-- 2s

00:03:57.170 --> 00:04:01.910
squared over g-- times-- now
negative sine of theta times

00:04:01.910 --> 00:04:03.800
sine of theta, that's
just negative

00:04:03.800 --> 00:04:06.790
sine squared of theta.

00:04:06.790 --> 00:04:10.110
And then, cosine theta times
cosine theta, that's just plus

00:04:10.110 --> 00:04:12.440
cosine squared of theta.

00:04:12.440 --> 00:04:16.339
Now, what we just said is we
want to figure out the point,

00:04:16.339 --> 00:04:19.170
the angle at which this
derivative or the

00:04:19.170 --> 00:04:21.620
instantaneous slope is 0.

00:04:21.620 --> 00:04:25.430
So let's set this thing
equal to 0.

00:04:25.430 --> 00:04:27.450
So we just have to solve
for theta now.

00:04:27.450 --> 00:04:33.760
Now the first thing I do to
solve for theta is just divide

00:04:33.760 --> 00:04:37.280
both sides by 2s
squared over g.

00:04:37.280 --> 00:04:39.900
If you divide the left-hand side
by that, it cancels out

00:04:39.900 --> 00:04:41.350
with 2s squared over g.

00:04:41.350 --> 00:04:45.040
And if you divide 0 by that,
assuming this isn't 0, which

00:04:45.040 --> 00:04:47.550
it shouldn't be, then
you'll still get 0.

00:04:47.550 --> 00:04:51.660
So this equation simplifies to--
I'll write it in blue--

00:04:51.660 --> 00:04:57.540
negative sine squared of theta
plus cosine squared of theta

00:04:57.540 --> 00:04:59.630
is equal to 0.

00:04:59.630 --> 00:05:03.520
Now, if we add sine squared of
theta of both sides of this

00:05:03.520 --> 00:05:06.580
equation, let's add
sine squared of

00:05:06.580 --> 00:05:09.660
theta to both sides.

00:05:09.660 --> 00:05:14.260
We are left with--
these cancel out.

00:05:14.260 --> 00:05:17.800
Cosine squared of theta
is equal to

00:05:17.800 --> 00:05:19.270
sine squared of theta.

00:05:22.470 --> 00:05:25.430
Now, both of these are going
to be positive over the

00:05:25.430 --> 00:05:27.470
interval, so we're going to just
take the positive square

00:05:27.470 --> 00:05:30.260
root of both of them, or the
principal root of both sides

00:05:30.260 --> 00:05:31.680
of this equation.

00:05:31.680 --> 00:05:32.990
So let's do that.

00:05:32.990 --> 00:05:35.300
So you take the principal
roots of both

00:05:35.300 --> 00:05:37.740
sides of this equation.

00:05:37.740 --> 00:05:38.510
You could do it that way.

00:05:38.510 --> 00:05:42.590
Actually, a more interesting way
than doing it that way, is

00:05:42.590 --> 00:05:45.830
to divide both sides of this
equation by cosine squared of

00:05:45.830 --> 00:05:51.200
theta assuming that it's not
equal to 0 over this interval.

00:05:51.200 --> 00:05:55.990
So cosine squared of theta.

00:05:55.990 --> 00:05:58.330
You could also do it using the
positive square root, the

00:05:58.330 --> 00:06:00.070
principal root, either
one will work.

00:06:00.070 --> 00:06:02.950
But this is interesting because
the left-hand side

00:06:02.950 --> 00:06:07.110
cancels out to 1, and the 1 will
be equal to-- what's sine

00:06:07.110 --> 00:06:09.270
squared over cosine
squared of theta?

00:06:09.270 --> 00:06:17.090
Well that's the same thing as
sine of theta over cosine of

00:06:17.090 --> 00:06:24.740
theta squared.

00:06:24.740 --> 00:06:27.300
You have a square divided
by another square.

00:06:27.300 --> 00:06:30.190
That's the same thing as the
numerator divided by the

00:06:30.190 --> 00:06:30.500
denominator.

00:06:30.500 --> 00:06:31.820
That whole thing squared.

00:06:31.820 --> 00:06:34.590
And what's sine of theta divided
by cosine of theta?

00:06:34.590 --> 00:06:36.270
Well that's just the
tangent of theta.

00:06:39.090 --> 00:06:45.670
So we have 1 is equal to the
tangent squared of theta.

00:06:45.670 --> 00:06:49.470
Or, we could take the positive
square root of both sides of

00:06:49.470 --> 00:06:50.380
this equation.

00:06:50.380 --> 00:06:54.100
Tangent is positive over the
interval from 0 to 90 degrees,

00:06:54.100 --> 00:06:55.710
so that's cool to do.

00:06:55.710 --> 00:06:58.250
So if you take the positive
square root of both sides, you

00:06:58.250 --> 00:07:00.800
get the positive square
root of 1 is 1.

00:07:00.800 --> 00:07:04.410
1 is equal to tangent
of theta.

00:07:04.410 --> 00:07:06.900
And then you take the inverse
tan of both sides or the arc

00:07:06.900 --> 00:07:12.400
tan of both sides and you
get the arc tan of

00:07:12.400 --> 00:07:16.510
1 is equal to theta.

00:07:16.510 --> 00:07:19.660
And this is just a very fancy
way of saying theta's the

00:07:19.660 --> 00:07:23.270
angle that if you were take
its tangent, you get 1.

00:07:23.270 --> 00:07:26.780
And you could use a calculator
to solve that or you might

00:07:26.780 --> 00:07:28.900
just know that by memory.

00:07:28.900 --> 00:07:36.390
This theta, the arc tangent
of 1 is 45 degrees.

00:07:36.390 --> 00:07:39.030
Or if you are dealing
in radians, it

00:07:39.030 --> 00:07:43.750
is pi over 4 radians.

00:07:43.750 --> 00:07:45.910
Either one of those
is going to work.

00:07:45.910 --> 00:07:48.960
So our optimal angle when we
shoot this thing is going to

00:07:48.960 --> 00:07:52.160
be at 45 degrees.

00:07:52.160 --> 00:07:55.280
Now, what is that optimal
distance going to be when we

00:07:55.280 --> 00:07:57.370
shoot it off at 45 degrees?

00:07:57.370 --> 00:08:00.380
Well, we can just go back
to our original formula.

00:08:00.380 --> 00:08:01.840
We just go back to our original

00:08:01.840 --> 00:08:02.990
formula that we derived.

00:08:02.990 --> 00:08:06.500
If we're shooting it off at 45
degrees, what is the sine of

00:08:06.500 --> 00:08:07.500
45 degrees?

00:08:07.500 --> 00:08:11.330
The sine of 45 degrees is
equal to the square

00:08:11.330 --> 00:08:13.030
root of 2 over 2.

00:08:13.030 --> 00:08:15.390
You could use a calculator for
that or maybe you know it from

00:08:15.390 --> 00:08:16.500
the unit circle.

00:08:16.500 --> 00:08:21.410
The cosine of 45 degrees is also
square root of 2 over 2.

00:08:21.410 --> 00:08:23.700
And if you'd actually just taken
the principal roots at

00:08:23.700 --> 00:08:26.300
this stage of the equation,
you'd have gotten that the

00:08:26.300 --> 00:08:29.130
cosine of theta has to equal
sine of theta over this

00:08:29.130 --> 00:08:31.940
interval, and that only
happens at 45 degrees.

00:08:31.940 --> 00:08:32.710
But given this.

00:08:32.710 --> 00:08:37.760
We can put this back into the
original expression right up

00:08:37.760 --> 00:08:40.020
here, our original function.

00:08:40.020 --> 00:08:43.240
So the optimal distance that
we are going to travel, so

00:08:43.240 --> 00:08:47.440
distance as a function-- the
distance we travel at 45

00:08:47.440 --> 00:08:54.020
degrees is going to be equal
to 2s squared over g times

00:08:54.020 --> 00:08:57.690
cosine of theta, which is
square root of 2 over 2.

00:08:57.690 --> 00:09:01.570
Cosine of 45 is square root of
2 over 2 time sine of theta,

00:09:01.570 --> 00:09:04.590
which is square root
of 2 over 2.

00:09:04.590 --> 00:09:07.440
Well what's the square root of
2 times the square root of 2?

00:09:07.440 --> 00:09:08.880
Well that's just 2.

00:09:08.880 --> 00:09:10.770
Let me simplify this.

00:09:10.770 --> 00:09:12.240
So the square root of 2
times the square root

00:09:12.240 --> 00:09:14.010
of 2, that is 2.

00:09:14.010 --> 00:09:16.330
This 2 cancels out
with that 2.

00:09:16.330 --> 00:09:19.600
And then, this 2 cancels
out with this 2.

00:09:19.600 --> 00:09:22.850
So then the optimal distance you
travel at 45 degrees, all

00:09:22.850 --> 00:09:28.220
we're left with is the
s squared over g.

00:09:28.220 --> 00:09:31.030
Assuming no air resistance, kind
of an ideal circumstance.

00:09:31.030 --> 00:09:34.410
No matter what planet you're
on, how fast you do it, the

00:09:34.410 --> 00:09:37.250
best angle is always 45 degrees
assuming no air

00:09:37.250 --> 00:09:37.980
resistance.

00:09:37.980 --> 00:09:41.010
And if you do it on that best
angle, you're going to travel

00:09:41.010 --> 00:09:44.080
s squared over g.

00:09:44.080 --> 00:09:47.880
Going back to the original
problem, if s is

00:09:47.880 --> 00:09:49.150
10 meters per second.

00:09:49.150 --> 00:09:51.130
Let's say s was 10 meters
per second.

00:09:54.690 --> 00:09:58.270
And let's say we're dealing with
a world where let's say,

00:09:58.270 --> 00:10:03.640
gravity is equal to 10 meters
per second squared, then

00:10:03.640 --> 00:10:13.630
according to what we've derived,
your optimal distance

00:10:13.630 --> 00:10:20.410
is going to be s squared-- so
it's going to be 100-- divided

00:10:20.410 --> 00:10:21.050
by gravity.

00:10:21.050 --> 00:10:22.530
It's going to be 10.

00:10:22.530 --> 00:10:25.080
And if you square meters per
second, you're going to get

00:10:25.080 --> 00:10:28.420
meters squared per second
squared divided by the

00:10:28.420 --> 00:10:32.200
acceleration of gravity, meters
per second squared.

00:10:32.200 --> 00:10:33.550
Second squared's cancel out.

00:10:33.550 --> 00:10:36.290
You have a meter squared
divided by meters.

00:10:36.290 --> 00:10:39.080
Your optimal distance
would be 10 meters.

00:10:39.080 --> 00:10:40.530
Pretty neat.