If the volume of a spherical ball is increasing at the rate of $$4\pi$$ cc/sec then the rate of increase of its radius (in cm/sec), when the volume is $$288\pi$$ cc is:

The volume $$V$$ of a sphere is given by the formula $$V = \frac{4}{3}\pi r^3$$, where $$r$$ is the radius. We are given that the volume is increasing at a rate of $$\frac{dV}{dt} = 4\pi$$ cc/sec. We need to find the rate of increase of the radius, $$\frac{dr}{dt}$$, when the volume is $$V = 288\pi$$ cc.

First, differentiate the volume formula with respect to time $$t$$. Using the chain rule, the derivative of $$V$$ with respect to $$t$$ is:

$$\frac{dV}{dt} = \frac{d}{dt}\left(\frac{4}{3}\pi r^3\right)$$

Since $$\frac{4}{3}\pi$$ is a constant, it can be factored out:

$$\frac{dV}{dt} = \frac{4}{3}\pi \cdot \frac{d}{dt}(r^3)$$

The derivative of $$r^3$$ with respect to $$t$$ is $$3r^2 \frac{dr}{dt}$$ by the chain rule. Substituting this in:

$$\frac{dV}{dt} = \frac{4}{3}\pi \cdot 3r^2 \frac{dr}{dt}$$

Simplify the expression:

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

We know $$\frac{dV}{dt} = 4\pi$$ cc/sec. Substitute this value:

$$4\pi = 4\pi r^2 \frac{dr}{dt}$$

Divide both sides by $$4\pi$$ (assuming $$\pi \neq 0$$):

$$\frac{4\pi}{4\pi} = r^2 \frac{dr}{dt}$$

$$1 = r^2 \frac{dr}{dt}$$

Therefore, solving for $$\frac{dr}{dt}$$:

$$\frac{dr}{dt} = \frac{1}{r^2}$$

Now, we need to find $$r$$ when $$V = 288\pi$$ cc. Using the volume formula:

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

Substitute $$V = 288\pi$$:

$$288\pi = \frac{4}{3}\pi r^3$$

Divide both sides by $$\pi$$:

$$288 = \frac{4}{3} r^3$$

Multiply both sides by 3 to eliminate the denominator:

$$288 \times 3 = 4 r^3$$

$$864 = 4 r^3$$

Divide both sides by 4:

$$\frac{864}{4} = r^3$$

$$216 = r^3$$

Take the cube root of both sides:

$$r = \sqrt[3]{216}$$

Since $$6 \times 6 \times 6 = 216$$, we have:

$$r = 6 \text{ cm}$$

Now substitute $$r = 6$$ into the expression for $$\frac{dr}{dt}$$:

$$\frac{dr}{dt} = \frac{1}{6^2} = \frac{1}{36}$$

Thus, the rate of increase of the radius is $$\frac{1}{36}$$ cm/sec.

Comparing with the options, $$\frac{1}{36}$$ corresponds to Option D.

Hence, the correct answer is Option D.