Recall that the volume V of a sphere of radius r can be computed using the formula. If the radius of a spherical balloon is increasing at a rate of 5 cm/min, how fast is its volume increasing when the radius is 10 cm?

Given that,

A sphere has a volume of V and a radius r

Volume of sphere cam be determine using the formula

V = 4/3 πr³

V = 4πr³ / 3.

If the radius of the sphere is increasing by

dr / dt = 5cm / min

How fast is the volume increasing when r = 10cm

dV / dt = ?

From V = 4πr³ / 3

We can calculate dV/dr

dV/dr = 12πr² / 3

dV/dr = 4πr²

Then,

We want to find dV/dt

Using chain rule

dV/dt = dV/dr * dr / dt

dV/dt = 4πr² * 5

dV/dt = 20πr²

So, at r = 10

dV/dt { 20π * 10²

dV/dt = 6283.19 cm/min

The rate at which the volume increase is 6283.19 cm/min