Question

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?

Answer 1

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