Examples on Related Rates (Section 3.9)
Example 1
Air is being pumped into a spherical balloon so that its volume increases at a rate of 100
cm
3
/
s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
Guidelines
1. Identify
Given information
: The rate of increase of the volume of air is 100 cm
3
/
s.
Unknown
: The rate of increase of the radius when the
diameter
is 50 cm.
Draw a diagram
.
2. Express these quantities mathematically
Let
V
be the volume of the balloon and let
r
be its radius.
The rate of increase of the volume with respect to time is the derivative
dV
dt
.
The rate of increase of the radius is
dr
dt
.
3. Restate the given and the unknown as follows
Given
dV
dt
= 100 cm
3
/
s.
Unknown
dr
dt
when 2
·
r
= 50 cm.
4. Connect
dV
dt
and
dr
dt
The volume of a sphere:
V
=
4
3
πr
3
.
To use the given information, we differentiate each side of the equation with respect to
t
(the Chain Rule):
5. Substitute the given information into the resulting equation and solve for the
unknown
dr
dt
1
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Example 2
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away
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 Fall '08
 Noohi
 Calculus, Trigraph

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