Question

The radius r of a sphere is increasing at a rate of 5 inches per minute.

Answer 1

Explanations:

The formula for calculating the volume of spheres is expressed as:

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

The rate of change of volume of the sphere is given as:

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

a) Given the following parameters:

`\begin{gathered} \text{radius }r\text{ =}8inches \\ (dr)/(dt)=5in\text{/min} \end{gathered}`

Substitute into the result to have:

`\begin{gathered} (dV)/(dt)=4\pi(8)^2\cdot5 \\ (dV)/(dt)=4\pi*64*5 \\ (dV)/(dt)=1280\pi in^3\text{/min} \end{gathered}`

b) If the radius is 37inches, the rate of change of the volume is given as:

`\begin{gathered} (dV)/(dt)=4\pi(37)^2\cdot5 \\ (dV)/(dt)=4\pi*1369*5 \\ (dV)/(dt)=27,380\pi in^3\text{/min} \end{gathered}`