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RELATED RATES
In a related rates problems, we model words problems with mathematical prob
lems and apply the chain rule to the mathematical models. Our strategy here is as
follows.
0. Draw a diagram if possible.
1. Introduce notation.
2. Write an equation that relates the various quantities of the problem.
3. Use the chain rule or implicit diﬀerentiation to diﬀerentiate both sides of the
equation.
Example 1.
Sand is being emptied from a hopper at the rate of 10 ft
3
/s. The
sand forms a conical pile whose height is always twice its radius. At what rate is
the radius of the pile increasing when its height is 5 ft?
Solution
Let
r
be the radius of the cone. Since the height
h
= 2
r
, we can see the
volume of the cone
V
=
1
3
πr
2
(2
r
) =
2
3
πr
3
.
We have
dV
dt
= 10 and want to ﬁnd
dr
dt
at
h
= 5 or
r
= 2
.
5. Using the chain rule,
we get
dV
dt
=
dV
dr
dr
dt
= 2
πr
2
dr
dt
.
Thus, the radius is increasing at the rate of
10
12
.
5
π
= 0
.
255ft/s
.
Example 2.
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
 Fall '09
 BenedictGross
 Chain Rule, The Chain Rule

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