Chapter 11.3, Problem 69E

a)

To determine

To evaluate: the derivative of V with respect to r .

Answer to Problem 69E

The derivative is $4\pi r^2$ .

Explanation of Solution

Given:

  $V\left(r\right)=\frac{4}{3}\pi r^3$

Calculation: -

  $\begin{array}{c}V\left(r\right)=\frac{4}{3}\pi r^3\\ \frac{d}{dr}V\left(r\right)=\frac{d}{dr}\left(\frac{4}{3}\pi r^3\right)\\ =\frac{4}{3}\pi \left(3r^2\right)\\ =4\pi r^2\end{array}$

Hence, the derivative is $4\pi r^2$ .

b)

To determine

To evaluate: the value of derivative of V when r is 4 inches.

Answer to Problem 69E

The value of derivative of V when r is 4 inches is $64inc{h}^2$ .

Explanation of Solution

Given:

  $V\left(r\right)=\frac{4}{3}\pi r^3$

Calculation: -

  $\begin{array}{c}V\left(r\right)=\frac{4}{3}\pi r^3\\ \frac{d}{dr}V\left(r\right)=\frac{d}{dr}\left(\frac{4}{3}\pi r^3\right)\\ =\frac{4}{3}\pi \left(3r^2\right)\\ =4\pi r^2\\ =4\pi {\left(4\right)}^2\\ =64\pi inc{h}^2\end{array}$

Hence, the value of derivative of V when r is 4 inches is $64inc{h}^2$ .

c)

To determine

To evaluate: the type of unit applied in part b.

Answer to Problem 69E

The unit of area is applied.

Explanation of Solution

Given:

  $V\left(r\right)=\frac{4}{3}\pi r^3$

Calculation: -

  $\begin{array}{c}V\left(r\right)=\frac{4}{3}\pi r^3\\ \frac{d}{dr}V\left(r\right)=\frac{d}{dr}\left(\frac{4}{3}\pi r^3\right)\\ =\frac{4}{3}\pi \left(3r^2\right)\\ =4\pi r^2\\ =4\pi {\left(4\right)}^2\\ =64\pi inc{h}^2\end{array}$

Hence, the value of derivative of V when r is 4 inches is $64inc{h}^2$ .

Now, the unit applied in the second part is inch² .

It is a unit of area.

It is applied because after differentiating the volume, area is received.