Chain Rule Application Questions
1. The temperature T of a metal rod decreases with respect to distance x from the rod’s left end,
at a rate of 3°C/cm. An ant, which is trying to get cooler, crawls along the rod from left to right
at a rate of 2 cm/s.
a) Write the given rates of change as derivatives. dT  3 C /cm, dx  2cm /s
o

dx


dt
b) Use the Chain Rule to find the rate of change of temperature, as felt by the ant, with respect
to time t. [-6 °C/s]

2. Water flows into a bathtub at a rate of 20 L/min, and the level rises 0.1 cm for each litre added.
a) Write the given rates of change as derivatives. Use V to denote volume, h to denote the
height of the water surface, and t to denote time. dV  20 L /min, dh  0.1 cm /L

dt
dV


b) Use the Chain Rule to find how fast the water level is rising. [2 cm/min]

3. The radius of a circle is increasing at a rate of 3 cm/s. Find the change of the area of the circle
with respect to time when the radius is 10 cm. [60 cm2/s]
4 3
r . If the radius is increasing at a rate of
3
0.03 m/s when the radius is 3 m, find the rate of change of volume with respect to time, at that
instant. [1.08π m3/s]
4. The volume of a spherical balloon of radius r is V 
 m. The distance fallen (in metres) after t seconds is
5. A skydiver jumps from an airplane at 3000
approximated by s = 5t2. As she falls, she experiences a change in air pressure p that will cause
dp
her ears to “pop” if the rate of change of pressure
exceeds 2 pressure units/s. Suppose that
dt
the change of pressure with respect to altitude (in metre) is 0.075 pressure units/m. At what
time will the skydiver’s ears pop? At what height will this occur? [2.67 s, 35.6 m]
