Chapter 11.3, Problem 70E

(a)

To determine

To find: the derivative of S with respect to r.

Answer to Problem 70E

The derivative of S with respect to r is $S^{\prime }\left(r\right)=8\pi r$ .

Explanation of Solution

Given:

  $S\left(r\right)=4\pi r^2$

Calculation:

Find the derivative of the surface area S with respect to r using the power rule of derivatives

  $\begin{array}{l}\frac{dr}{dS}=\frac{d}{dS}\left[4\pi r^2\right]\\ =4\pi \left(2\right)r^{2-1}\\ =8\pi r\end{array}$

Conclusion:

Therefore,the derivative of S with respect to r is $S^{\prime }\left(r\right)=8\pi r$ .

(b)

To determine

To evaluate: the derivative when the radius is 3 millimeters.

Answer to Problem 70E

The derivative when the radius is 3 millimeters is $S^{\prime }\left(3\right)=24\pi \approx 75.398\frac{m{m}^2}{s}$ .

Explanation of Solution

Given:

Calculation:

Part B:

Substitute r for 3 to evaluate the rate of change of surface area of the tumor when its radius is 3 mm.

  $\begin{array}{l}{S}^{\prime }\left(3\right)=8\pi \left(3\right)\\ =24\pi \\ \approx 75.398\frac{m{m}^2}{s}\end{array}$

Conclusion:

Therefore,the derivative when the radius is 3 millimeters is $S^{\prime }\left(3\right)=24\pi \approx 75.398\frac{m{m}^2}{s}$ .

(c)

To determine

To explain: the type of unit would be applied to your answer.

Answer to Problem 70E

The derivative of surface area describes the rate of change of that surface area, meaning our units are millimeters squared per second.

Explanation of Solution

Calculation:

Because the derivative expresses slope at a given input value, it describes the rate of change of the original equation at the given input. Thus, the derivative of surface area describes the rate of change of that surface area, meaning our units are millimeters squared per second.

  $\begin{array}{l}\text{Rate of SA}\\ \text{reduction of the tumor = }75.398\frac{m{m}^2}{s}\end{array}$

Conclusion:

Therefore,the derivative of surface area describes the rate of change of that surface area, meaning our units are millimeters squared per second.