Question

The radius r of a sphere is increasing at a rate of 4 inches per minute. Find the rate of change of the volume when r =2 inches

Answer 1

Given the word problem, we can deduce the following information:

1. The radius r of a sphere is increasing at a rate of 4 inches per minute.

2. r =2 inches

To determine the change in volume, we first use the formula of the volume of sphere:

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

where:

V=Volume

r=radius

Next, we take the derivative of the volume:

$\begin{gathered} V=\frac{4}{3}\pi r^{3} \\ \frac{dV}{dt}=\frac{4}{3}\frac{\pi3r^2dr}{dt} \\ Simplify \\ \frac{dV}{dt}=4\pi r^2\cdot\frac{dr}{dt} \\ \end{gathered}$

Hence,

$\frac{dr}{dt}=4\frac{\text{ inches}}{minute}$$r=2\text{ inches}$

We plug in r=2 and dr/dt=4 into the derivative of the volume of sphere:

$\begin{gathered} \frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt} \\ \frac{dV}{dt}=4\pi(2)^2(4) \\ Calculate \\ \frac{dV}{dt}=201.06\text{ }\frac{in^3}{minute} \end{gathered}$

Therefore, the rate of change of the volume is 201.06 in^3/min.