Chapter 11.3, Problem 70E

a)

To determine

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

Answer to Problem 70E

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

Explanation of Solution

Given:

  $S\left(r\right)=4\pi r^2$

Calculation: -

  $\begin{array}{c}S\left(r\right)=4\pi r^2\\ \frac{d}{dr}S\left(r\right)=\frac{d}{dr}\left(4\pi r^2\right)\\ =4\pi \left(2r\right)\\ =8\pi r\end{array}$

Hence, the derivative is $8\pi r$ .

b)

To determine

To evaluate: the value of derivative of S when r is 3 mm.

Answer to Problem 70E

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

Explanation of Solution

Given:

  $S\left(r\right)=4\pi r^2$

Calculation: -

  $\begin{array}{c}S\left(r\right)=4\pi r^2\\ \frac{d}{dr}S\left(r\right)=\frac{d}{dr}\left(4\pi r^2\right)\\ =4\pi \left(2r\right)\\ =8\pi r\\ =8\pi \left(3\right)\\ =24\pi \end{array}$

Hence, the value of derivative of S when r is 3 mm is $24\pi m{m}^2$ .

c)

To determine

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

Answer to Problem 70E

The unit of area is applied.

Explanation of Solution

Given:

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

Calculation: -

  $\begin{array}{c}S\left(r\right)=4\pi r^2\\ \frac{d}{dr}S\left(r\right)=\frac{d}{dr}\left(4\pi r^2\right)\\ =4\pi \left(2r\right)\\ =8\pi r\\ =8\pi \left(3\right)\\ =24\pi \end{array}$

Hence, the value of derivative of S when r is 3 mm is $24\pi m{m}^2$ .

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

It is a unit of length.

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