AP Calculus AB Exam Question:

A spherical balloon is being inflated at a constant rate of 3 cubic inches per second. Determine the rate at which the radius of the balloon is changing when the radius is 4 inches.

Answer:

To solve this related rates problem, we need to find the relationship between the radius of the balloon and its volume, and then differentiate both sides with respect to time.

Let's denote the radius of the balloon as r (in inches) and the volume of the balloon as V (in cubic inches).

The volume of a sphere can be given by the formula: V = (4/3) * π * r^3.

Given that the balloon is being inflated at a constant rate of 3 cubic inches per second, we can express the change in volume over time as dV/dt = 3 (in cubic inches per second).

We are trying to find the rate at which the radius of the balloon is changing, dr/dt, when the radius is 4 inches.

To find this rate, we'll differentiate the volume formula with respect to time:

dV/dt = (d/dt)[(4/3) * π * r^3].

Differentiating both sides, we have:

3 = (4/3) * π * 3r^2 * (dr/dt).

Now, let's solve for dr/dt:

dr/dt = (3 * 3) / [(4/3) * π * (4^2)].

Simplifying further:

dr/dt = 9 / [(4/3) * 16 * π].

dr/dt = 9 / (64/3 * π).

dr/dt = 27 / (64 * π).

So, the rate at which the radius of the balloon is changing when the radius is 4 inches is 27 / (64 * π) inches per second.