Question

The radius r of a sphere is increasing at a rate of 6 inches per minute. (a) find the rate of change of the volume when r = 10 inches.

Answer 1

Given:
Increasing rate of radius (r) of a sphere
dr/dt -> 6inches per min
And Radius (r) of sphere —> 10 inches

Sol:
Vol. of sphere => V = 4/3 πr^3

Derivative of V respect to time (t) is:
dV/dt = 4/3 π d/dt (r^3)
= 4/3 π (3r^2 dr/dt)
=4πr^2 dr/dt
= 4π(10)^2 (6) (Since, dr/dt is 6 & r= 10)
= 2400π


Therefore,
The increasing change in vol. of the sphere is 2400π cubic inches per min.

Hope this helps.