1
00:00:00,012 --> 00:00:13,201
[music].
>> I'm going to blow up this balloon.

2
00:00:13,202 --> 00:00:22,978
Let's suppose that I'm blowing air into
that balloon at a constant rate, so the

3
00:00:22,978 --> 00:00:28,911
balloon's volume is changing at a constant
rate.

4
00:00:28,911 --> 00:00:34,013
How quickly is the radius of the balloon
changing?

5
00:00:34,014 --> 00:00:39,369
We can model the balloon as a sphere.
And recall the volume of a sphere in terms

6
00:00:39,369 --> 00:00:42,989
of its radius r.
The volume of a sphere of radius r is 4

7
00:00:42,989 --> 00:00:46,685
3rds pi r cubed.
So how fast does the radius change as I

8
00:00:46,685 --> 00:00:50,898
blow the balloon up?
So I've got the original equation for the

9
00:00:50,898 --> 00:00:55,714
volume of a sphere of radius r.
I'm going to differentiate both sides with

10
00:00:55,714 --> 00:00:59,972
respect to time.
So the derivative of v, with respect to

11
00:00:59,972 --> 00:01:05,346
time, is I'm going to write dv dt.
And the derivative of the other side, with

12
00:01:05,346 --> 00:01:09,698
respect to time, we've gotta think a
little bit more about.

13
00:01:09,698 --> 00:01:15,417
Now 4 3rds and pi are just constants.
So I can pull them out of the derivative.

14
00:01:15,418 --> 00:01:20,337
But I've still gotta figure out what the
time derivative of r cubed is.

15
00:01:20,337 --> 00:01:25,635
To figure out the time derivative of r
cubed, I'm regarding r, remember, as a

16
00:01:25,635 --> 00:01:29,143
function of t.
So I use the chain rule, and the

17
00:01:29,143 --> 00:01:36,073
derivative of cubing is 3 times the inside
function squared times the derivative of

18
00:01:36,073 --> 00:01:42,132
the inside function, so dr dt.
So dv dt is 4 3rds pi times 3 times r

19
00:01:42,132 --> 00:01:48,377
squared times dr dt.
Let's solve for dr dt, the rate of change

20
00:01:48,377 --> 00:01:53,535
in the balloon's radius.
All right, so I'm going to solve for dr

21
00:01:53,535 --> 00:01:56,493
dt.
Well, first thing I can do is, I've got a

22
00:01:56,493 --> 00:02:01,299
dividing by 3, and a multiplying by 3.
So I don't need to do both of those

23
00:02:01,299 --> 00:02:04,781
operations.
So this is 4 pi r squared times dr dt.

24
00:02:04,781 --> 00:02:09,026
The other side is just dv dt.
So I'll divide both sides by 4 pi r

25
00:02:09,026 --> 00:02:12,843
squared.
And I'll find out that, dr dt is dv dt

26
00:02:12,843 --> 00:02:18,341
divided by 4 pi r squared.
Can we really make use of this equation?

27
00:02:18,341 --> 00:02:22,664
I, does this help us at all?
Yeah, this makes sense.

28
00:02:22,664 --> 00:02:29,170
We started with the original equation for
the volume of a sphere in terms of its

29
00:02:29,170 --> 00:02:33,040
radius.
And I differentitated this equation.

30
00:02:33,040 --> 00:02:37,802
And then solved for dr dt.
And what this equation is telling me is

31
00:02:37,802 --> 00:02:42,977
that the change in radius, which is what
dr dt is measuring, the rate of change in

32
00:02:42,977 --> 00:02:47,927
the size of the balloon, well, it depends
on how quickly the volume's changing,

33
00:02:47,927 --> 00:02:51,036
which is dv dt.
But it also depends upon the radius,

34
00:02:51,036 --> 00:02:54,741
right?
This is this r squared in the denominator.

35
00:02:54,741 --> 00:03:00,379
In particular, a really big balloon
doesn't get a whole lot bigger for a given

36
00:03:00,379 --> 00:03:05,835
change in volume, compared to a little
tiny balloon, which gets quite a bit

37
00:03:05,835 --> 00:03:11,882
bigger for the same change in volume.
It all has to do with that r squared in

38
00:03:11,882 --> 00:03:16,672
the denominator.
When the balloon's radius is very small, a

39
00:03:16,672 --> 00:03:21,253
given change in volume results in quite a
dramatic change in the radius.

40
00:03:21,253 --> 00:03:26,219
But once the radius is large, the r
squared in the denominator means that dr

41
00:03:26,219 --> 00:03:29,828
dt is then much smaller for the same
change in volume.

42
00:03:31,153 --> 00:03:40,945
Another breath, and yeah, I mean, the
thing's still getting bigger, but much

43
00:03:40,945 --> 00:03:45,379
less dramatically.
.