Question

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.

Answer 1

Answer:

0.28 cm

Explanation:

The volume of a sphere is given by:

$V=\frac{4}{3}\pi r^3$

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

The rate of change of the volume is

$\frac{dV}{dt}=4 \pi r^2 \frac{dr}{dt}$ (1)

where

$\frac{dr}{dt}$ is the rate of change of the radius. We know that

$\frac{dr}{dt}=9$ (cm/s)

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

$\frac{dV}{dt}=9$ (cm^3/s)

So we can rewrite (1) as:

$9=4\pi r^2 \cdot 9$

By solving it, we find

$4\pi r^2 = 1\\r = \sqrt{\frac{1}{4\pi}}=0.28 cm$