Final answer:
The rate of change of the radius of a spherical balloon being pumped with air at the rate of 4.5 cubic inches per minute, when the radius is 2 inches, is calculated using related rates and differentiating the volume formula of a sphere. It is found to be 4.5/16π inches per minute.
Step-by-step explanation:
To find the rate of change of the radius as air is being pumped into a spherical balloon, we use related rates, which involves calculus, and the formula for the volume of a sphere given by V = \frac{4}{3}\pi r^3.
Differentiating both sides of the volume equation with respect to time t, we get dV/dt = 4 \pi r^2 dr/dt, where dV/dt is the rate of change of the volume and dr/dt is the rate of change of the radius.
Given that dV/dt = 4.5 cubic inches per minute and the radius r is 2 inches, we can plug these values into the differentiated equation and solve for dr/dt:
4.5 = 4 \pi (2)^2 dr/dt
4.5 = 16 \pi dr/dt
dr/dt = \frac{4.5}{16\pi}
The rate of change of the radius when the radius is 2 inches is \frac{4.5}{16\pi} inches per minute.