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=(4)/(3)\pi r^3`

where:

V=Volume

r=radius

Next, we take the derivative of the volume:

`\begin{gathered} V=(4)/(3)\pi r^(3) \\ (dV)/(dt)=(4)/(3)(\pi3r^2dr)/(dt) \\ Simplify \\ (dV)/(dt)=4\pi r^2\cdot(dr)/(dt) \\ \end{gathered}`

Hence,

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

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