The radius of a sphere is increasing at a rate of 9 cm/ sec. find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.

0.28 cm

Explanation:

The volume of a sphere is given by:

where r is the radius, which is dependent on the time, so r(t).

The rate of change of the volume is

(1)

where

is the rate of change of the radius. We know that

(cm/s)

And we want to find the value of the radius r when the rate of change of the volume is the same:

(cm^3/s)

So we can rewrite (1) as:

By solving it, we find

e/c, where e is its total energy