Rates of Change and Related Rates: Areas and Volumes
1. The formula for the volume of a sphere is V =
4 3
r . What is the
3
change in volume as the radius changes from 3cm to 6cm? 252π
2. Water is being pumped from a tank so that the volume V in litres
remaining after t minutes is determined by V(t) = 1000(10 – t2) for 0<
t < 10 . Over the first 2 minutes, what is the average rate of change
of the volume remaining?-2000
3. A designer is experimenting with a cylindrical can with a fixed height
of 15 cm. Find the rate of change of volume with respect to radius
when the radius is 4cm. Vcylinder = r 2h . (120π)
4. Find the rate of change of the area A of a square with respect to its
side length s. (2s)
5. Find the rate of change of the area A of a square with respect to its
diagonal length d. (d)
6. The local newspaper purchases its newsprint in large rolls 1.25m in
diameter. If the cross sectional area of a roll decreases at a
constant rate of 0.5m2 per hour when the presses are rolling, how fast
is the radius of the roll decreasing when the radius is 0.75m? 1/(3π)
7. Gas is escaping from a spherical weather balloon at a rate of
50cm3/min. How fast is the surface area S shrinking when the radius
is 15m? (1/15cm2/min)
8. Water is pouring into an inverted right circular cone at a rate of 
m3/min. The height and the diameter of the base of the cone are
both 10m. How fast is the water level rising when its depth is 8m?
(1/16)
9. Each edge of a cube is expanding at a rate of 4cm/s.
a. How fast is the volume changing when each edge is 5 cm? (300)
b. At what rate is the surface area changing when each edge is
7cm? (336)
10. A spherical balloon is being filled (Vsphere =
4 3
r ) with helium at a rate
3
of 8cm3/s. At what rate is its radius increasing
a. When the radius is 12 cm? (1/72π)
b. When the volume is 1435 cm3? (0.013)
c. When it has been filling for 33.5 s? (0.0397)
11. A cylindrical tank with height 15m and diameter 2m is being filled with
gasoline at a rate of 500L/min. At what rate is the fluid level in the
tank rising? (1L = 1000cm3) About how long will it take to fill the tank?
(1/2π).
12. Suppose that a raindrop is a perfect sphere. Assume that through
condensation the raindrop accumulates moisture at a rate proportional
to its surface area. Show that the radius increases at a constant rate.
White
1. p191 #2
2. p191 #4
3. p191 #6
4. p191 #11
5. p191 #13
6. p191 #14
Yellow:
1. p191 example #1
2. p191 #4
3. p191 #12
4. p191 #13
5. p127 #16