A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.2 cm/min

Question

. At what rate is the volume of the snowball decreasing when the diameter is 15 cm. (Note the answer is a positive number).

Answer 1

We generally know the following formula for the volume V of a sphere of radius R,

V = (4/3)πR³

But, this problem is expressed in terms of the diameter D = 2R. Or, R = D/2.

Therefore, we can write

V = (4/3)π(D/2)³

V = (4/3)π(D³/2³)

V = (4/3)π(D³/8)

V = ((π/6)D³

Take the time derivative.

dV/dt = (π/6)3D² dD/dt using the Chain Rule

dV/dt = (π/2) D² dD/dt

We are told that the diameter D decreases at a rate of 0.2 cm/min, so dD/dt = 0.2 cm/min

Let D = 15 cm.

Then, dV/dt = (π/2) (15 cm)² (0.2 cm/min)

dV/dt = 22.5 π cm³/min