Radius of a sphere

Question

The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing (in mm³/s) when the diameter is 80 mm? (Round your answer to two decimal places.)

Answer 1

Let dr/dt be the rate of change of the sphere's radius (r), with respect to time (t).

The volume of a sphere (V) as a function of the radius is (4π÷3)r³. The rate of change of the volume with respect to the radius dV/dr is

dV/dr = (4π÷3)(3r²) = 4πr².

Since the sphere's radius is a function of time and the volume is a function of the radius, the sphere's volume can also be expressed as a function of time dV/dt.

dV/dt = dV/dr × dr/dt

We are told that the radius is increasing at 3 mm/s, so dr/dt = 3 mm/s

We can calculate dV/dr when the diameter is 80 mm (r = 40 mm)

dV/dr = (4π÷3)(3r²) = 4πr² = 4π(40)² mm².

Finally, we can calculate the rate of change of the volume:

dV/dt = dV/dr × dr/dt = [4π(40)² mm²]×[3 mm/s] = 6.0 × 10⁴ mm³/s