Mth 201
Related Rates Practice Problems
1. A spherical snowball is melting. Its radius is decreasing at 0.2 cen-
timeters per hour when the radius is 15cm. How fast is the volume
decreasing at this time?
Solution:
•Variables Volume: V, radius: rand time: t
•We want dV/dt - rate of change of volume with respect to time.
•We know dr/dt =−0.2 - the rate of change of radius with respect
to time (it is negative since the radius is decreasing).
•Equations relating variables: V= 4πr3/3 (volume of a sphere in
terms of radius).
•Solving the problem: We want dV/dt, so we need to differentiate
both sides with respect to t. Differentiating, we get
dV
dt = 4πr2dr
dt .
Plugging in r= 15 and dr/dt =−0.2, we get dv/dt = 4π(15)2(−0.2) ∼
−565.5cm3/hr.
2. A train is traveling at 0.8km/min along a straight track, moving in the
direction as shown below. A movie camera, 0.5km away from the track
is focused on the train.
Train
Camera
0.8kn/hr
Theta z
x
0.5km
(a) Express z, the distance between the camera and the train as a
function of x.
Solution: Using Pythagoras Theorem, we have x2+ (0.5)2=z2,
so
z=rx2+1
4.
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