The radius of a sphere is increasing at a rate of 3 mm/s. how fast is the volume of the sphere increasing when the diameter is 10 mm?

We will find the rate of change of volume with respect to the change of radius.

Volume, V (r) = (4/3) π r^3

Find rate of change of volume with respect to change of radius by differentiation

V' (r) = dV/dr = d[ (4/3) π r^3]/dr = (4/3) π d[r^3]/dr = (4/3) π 3r^2

Therefore when diameter=10mm, r=5mm

V' (r) = V' (5) = (4/3) π (3 (5^2)) = (4/3) π 75 = 100 π mm^3/mm