WEBVTT

00:00:00.000 --> 00:00:07.740
PROFESSOR: Hi everybody.

00:00:07.740 --> 00:00:09.280
Welcome back to recitation.

00:00:09.280 --> 00:00:12.100
In lecture you did a bunch
of examples of related rates

00:00:12.100 --> 00:00:13.040
problems.

00:00:13.040 --> 00:00:16.550
So I have a couple more
for you to do today.

00:00:16.550 --> 00:00:19.790
So here we've got--
OK, so we've got

00:00:19.790 --> 00:00:23.110
air being blown into
a spherical balloon

00:00:23.110 --> 00:00:26.690
at a rate of 1,000 cubic
centimeters per second.

00:00:26.690 --> 00:00:29.170
So the question is,
how fast is the radius

00:00:29.170 --> 00:00:32.799
growing when the radius
is equal to 8 centimeters?

00:00:32.799 --> 00:00:34.840
And then OK, so I've got
a second question, which

00:00:34.840 --> 00:00:38.810
is, how fast is the surface
area growing at that same time?

00:00:38.810 --> 00:00:41.250
So why don't you take
a few minutes, work

00:00:41.250 --> 00:00:42.849
this one out for
yourself, come back,

00:00:42.849 --> 00:00:44.140
and we'll work it out together.

00:00:51.520 --> 00:00:52.020
All right.

00:00:52.020 --> 00:00:52.830
Welcome back.

00:00:52.830 --> 00:00:57.520
So this question, like all
related rates questions,

00:00:57.520 --> 00:00:59.640
has the property that
the calculus is typically

00:00:59.640 --> 00:01:01.890
very straightforward,
but that there's

00:01:01.890 --> 00:01:05.040
some geometric or
algebraic setup.

00:01:05.040 --> 00:01:08.410
So in this case, it's
straight up geometry.

00:01:08.410 --> 00:01:13.300
So we have a balloon, we
know it's a perfect sphere,

00:01:13.300 --> 00:01:16.540
we know how fast the
volume is changing.

00:01:16.540 --> 00:01:19.050
So OK, but we need to know how
fast the radius is changing.

00:01:19.050 --> 00:01:21.180
So in order to do that
we need to figure out

00:01:21.180 --> 00:01:23.910
a relationship between
the radius and the volume.

00:01:23.910 --> 00:01:26.340
And then we can just do
implicit differentiation

00:01:26.340 --> 00:01:27.540
like we've been doing.

00:01:27.540 --> 00:01:31.580
So for example, OK,
so for a sphere--

00:01:31.580 --> 00:01:34.020
the setup is not so bad
for this first one--

00:01:34.020 --> 00:01:42.170
so we know that for a sphere the
volume is equal to 4/3 pi times

00:01:42.170 --> 00:01:43.670
the radius cubed.

00:01:43.670 --> 00:01:47.980
So that's the fundamental
relationship between the volume

00:01:47.980 --> 00:01:50.410
and radius of a sphere, and
it's true for every sphere

00:01:50.410 --> 00:01:52.920
everywhere in Euclidean space.

00:01:52.920 --> 00:01:56.880
And we're given
also, that the volume

00:01:56.880 --> 00:02:03.121
is changing at a constant rate
of 1,000 centimeters cubed

00:02:03.121 --> 00:02:03.620
per second.

00:02:03.620 --> 00:02:07.470
So dV/dt is just
given to be 1,000.

00:02:07.470 --> 00:02:10.650
You know, leave off the
units at this point.

00:02:10.650 --> 00:02:13.135
So the question
is, what is dr/dt?

00:02:13.135 --> 00:02:14.760
That's what we're
trying to figure out.

00:02:14.760 --> 00:02:16.810
We're trying to figure
out how fast the radius is

00:02:16.810 --> 00:02:19.810
changing at the moment
when the radius is

00:02:19.810 --> 00:02:21.260
equal to 8 centimeters.

00:02:21.260 --> 00:02:22.600
So how can we do that?

00:02:22.600 --> 00:02:25.790
Well, this fundamental
relationship, it's an identity.

00:02:25.790 --> 00:02:26.850
It always holds.

00:02:26.850 --> 00:02:30.020
So that means we can
differentiate it.

00:02:30.020 --> 00:02:33.400
So if we take the
derivative of this identity,

00:02:33.400 --> 00:02:40.110
well, V on the left
just becomes dV/dt.

00:02:40.110 --> 00:02:43.880
And on the right we want to
do implicit differentiation.

00:02:43.880 --> 00:02:46.930
So here r is changing
with respect to time.

00:02:46.930 --> 00:02:47.980
r is a function of t.

00:02:47.980 --> 00:02:48.480
Well, OK.

00:02:48.480 --> 00:02:50.500
So 4/3 pi is a constant.

00:02:50.500 --> 00:02:53.520
So that when we differentiate
nothing happens.

00:02:53.520 --> 00:02:54.560
4/3 pi.

00:02:54.560 --> 00:02:58.180
And so now we differentiate
r cubed with respect to t,

00:02:58.180 --> 00:03:07.160
so that gives us 3 r
squared times dr/dt.

00:03:07.160 --> 00:03:09.790
So that's just the chain
rule in action there.

00:03:09.790 --> 00:03:13.040
And now what we
want is this dr/dt.

00:03:13.040 --> 00:03:14.550
Right?

00:03:14.550 --> 00:03:17.000
That's the thing that
we're looking for, is

00:03:17.000 --> 00:03:18.760
how fast the radius is growing.

00:03:18.760 --> 00:03:20.310
So that's dr/dt.

00:03:20.310 --> 00:03:23.800
And we want it at the
moment when r is equal to 8.

00:03:23.800 --> 00:03:32.160
So when r is equal to 8,
this implies that-- well, OK,

00:03:32.160 --> 00:03:38.220
so dV/dt is 1,000 always.

00:03:38.220 --> 00:03:40.550
And it implies that--
OK, so it's equal to,

00:03:40.550 --> 00:03:53.410
1,000 is equal to 4/3 pi times
3 times 8 squared times dr/dt.

00:03:53.410 --> 00:03:58.000
So at this moment that
we're interested in,

00:03:58.000 --> 00:04:01.026
we have this equation
to solve, for dr/dt,

00:04:01.026 --> 00:04:02.900
and this is a nice,
simple equation to solve.

00:04:02.900 --> 00:04:04.480
You just divide
through by everything

00:04:04.480 --> 00:04:06.990
on the right-hand
side other than dr/dt.

00:04:06.990 --> 00:04:13.320
So this implies that dr/dt
is equal to-- well, OK,

00:04:13.320 --> 00:04:16.000
so I have to divide
1,000 by all this stuff.

00:04:16.000 --> 00:04:21.096
I think it works out to
something like 125 over 32*pi.

00:04:21.096 --> 00:04:23.360
All right.

00:04:23.360 --> 00:04:24.881
So that's the exact value.

00:04:24.881 --> 00:04:26.630
Maybe you're interested
in sort of knowing

00:04:26.630 --> 00:04:28.670
about how large this is.

00:04:28.670 --> 00:04:35.687
So 32*pi is pretty close to 100,
so this is about 1.2 something.

00:04:35.687 --> 00:04:36.270
So, all right.

00:04:36.270 --> 00:04:36.900
So there we go.

00:04:36.900 --> 00:04:38.530
So that answers
the first question.

00:04:38.530 --> 00:04:44.160
At that moment the radius is
growing at a rate of 125 over

00:04:44.160 --> 00:04:46.820
32*pi centimeters per second.

00:04:46.820 --> 00:04:47.320
OK.

00:04:47.320 --> 00:04:49.240
So now how about the second
question that we've got here?

00:04:49.240 --> 00:04:50.610
What about the surface area?

00:04:50.610 --> 00:04:54.707
So again, we know how fast
now the radius is changing

00:04:54.707 --> 00:04:56.540
and we know how fast
the volume is changing.

00:04:56.540 --> 00:04:58.900
So in order to figure out
how fast the surface area is

00:04:58.900 --> 00:05:01.670
changing, we need something
that relates the surface

00:05:01.670 --> 00:05:04.300
area to either the
volume or the radius.

00:05:04.300 --> 00:05:07.230
Now the relationship between
surface area and volume

00:05:07.230 --> 00:05:09.780
is something that we could
sort of work out if we had to,

00:05:09.780 --> 00:05:11.738
but it's a lot easier to
write down the surface

00:05:11.738 --> 00:05:12.910
area in terms of the radius.

00:05:12.910 --> 00:05:13.790
So let's do that.

00:05:16.920 --> 00:05:19.270
So we have, I'm going to
use the letter S to denote

00:05:19.270 --> 00:05:20.930
surface area of a sphere.

00:05:20.930 --> 00:05:23.540
So again, it's a general
identity, you know,

00:05:23.540 --> 00:05:27.480
a geometric fact that the
surface area of a sphere is

00:05:27.480 --> 00:05:31.960
equal to 4*pi times
the radius squared.

00:05:31.960 --> 00:05:34.000
And this is always true.

00:05:34.000 --> 00:05:36.570
And now the thing that
we want is the rate

00:05:36.570 --> 00:05:38.400
of change of the surface area.

00:05:38.400 --> 00:05:40.850
So the rate of change
is the derivative.

00:05:40.850 --> 00:05:45.570
So we want to compute the
derivative here, dS/dt.

00:05:45.570 --> 00:05:46.580
So OK, so we just do it.

00:05:46.580 --> 00:05:52.500
So dS/dt is equal to--
well, 4*pi hangs around.

00:05:52.500 --> 00:05:55.180
And again, we
differentiate r squared.

00:05:55.180 --> 00:05:58.520
r is a function of t, so we
have to use the chain rule here.

00:05:58.520 --> 00:06:06.270
So this is times 2r times dr/dt.

00:06:06.270 --> 00:06:07.890
So this is an identity.

00:06:07.890 --> 00:06:09.510
So this is true always.

00:06:09.510 --> 00:06:13.120
And now we want to know, at
this particular moment in time,

00:06:13.120 --> 00:06:16.055
when r is equal to
8, what is ds/dt?

00:06:16.055 --> 00:06:18.610
And in order figure that
out, well OK, we just

00:06:18.610 --> 00:06:22.350
have to be able to plug
in for everything else.

00:06:22.350 --> 00:06:26.420
So when r equals 8--
all right, well luckily,

00:06:26.420 --> 00:06:29.220
you know, if we were just
starting this problem

00:06:29.220 --> 00:06:31.250
from scratch here,
we'd have a problem.

00:06:31.250 --> 00:06:33.810
Which is we wouldn't
know what dr/dt was.

00:06:33.810 --> 00:06:36.455
But luckily, we've already
figured it out, right?

00:06:36.455 --> 00:06:37.830
In the first part
of the problem.

00:06:37.830 --> 00:06:42.386
So we know that when
r is equal to 8,

00:06:42.386 --> 00:06:49.550
dr/dt is equal to
125 over 32*pi.

00:06:49.550 --> 00:06:50.750
Did I copy that right?

00:06:50.750 --> 00:06:51.490
Yes, I did.

00:06:51.490 --> 00:06:51.990
OK.

00:06:55.610 --> 00:06:58.722
So OK, so in this case, the
equation we have to solve

00:06:58.722 --> 00:07:00.180
is just completely
straightforward.

00:07:00.180 --> 00:07:03.400
We just plug in the values and
it's already solved for us.

00:07:03.400 --> 00:07:04.760
So that's nice.

00:07:04.760 --> 00:07:12.660
So that we get that ds/dt at
that moment is equal to-- well,

00:07:12.660 --> 00:07:28.610
it's 4*pi times 2 times 8 is
the radius times 125 over 32*pi.

00:07:28.610 --> 00:07:29.625
Oh boy, and all right.

00:07:29.625 --> 00:07:32.080
So we can work this out
if we want, I guess.

00:07:32.080 --> 00:07:36.980
That's 32*pi to cancel,
so that's equal to 250.

00:07:36.980 --> 00:07:40.070
And I guess the units there
better be centimeters squared

00:07:40.070 --> 00:07:41.060
per second.

00:07:41.060 --> 00:07:43.680
So at this moment in
time the surface area

00:07:43.680 --> 00:07:47.130
is growing by 250 centimeters
squared per second.

00:07:47.130 --> 00:07:50.140
So that's all we had to do.

00:07:50.140 --> 00:07:51.860
So we're all set.