A spherical balloon is inflated with gas at a rate of 800 cubic centimeters per minute. (a) How fast is the radius of the balloon changing at the instant the radius is 70 centimeters

0.013 cm/minute

Explanation:

We are given;

Volume rate; dV/dt = 800 cm³/min

Now, volume of a sphere is given by the formula;

V = (4/3)πr³

We want to find the rate at which the radius of the balloon is changing. This is represented by dr/dt.

Now, dr/dt will be gotten from the relation;

dr/dt = dV/dt ÷ dV/dr

Now, dV/dr = 3 × (4/3)πr²

dV/dr = 4πr²

Thus;

dr/dt = 800/(4πr²)

At r = 70 cm

dr/dt = 800/(4π × 70²)

dr/dt = 0.013 cm/minute