What is the rate of increase of the radius of a spherical ball whose volume is increasing at the rate of 4π cc/sec when the volume is 288π cc?

Understanding the Problem:

We need to find the rate of increase of the radius of a spherical ball whose volume is growing at a constant rate. Given are:

• Volume increase rate: 4π cc/sec
• Volume at the point of interest: 288π cc

Volume of a Sphere:

The volume V of a sphere is given by the formula:

V = (4/3)πr³

where r is the radius of the sphere.

Differentiate the Volume Formula:

To find the rate at which the radius is changing with respect to time, we differentiate volume with respect to time:

dV/dt = 4πr²(dr/dt)

Given Data:

From the problem:

• dV/dt = 4π

Find the Radius when Volume is 288π:

We find the radius when the volume is 288π:

(4/3)πr³ = 288π

Dividing both sides by π, we get:

(4/3)r³ = 288

Now multiplying by 3 and dividing by 4:

r³ = 216

Taking the cube root gives us:

r = 6 cm

Substituting Values:

Now substitute back into the differentiated equation:

4π = 4π(6²)(dr/dt)

Calculating dr/dt:

Simplifying:

4 = 36(dr/dt)

Solving for dr/dt gives:

dr/dt = 4/36 = 1/9 cm/sec

Conclusion:

The rate at which the radius of the spherical ball is increasing when the volume is 288π cc is 1/9 cm/sec.