Chapter 11.3, Problem 69E

(a)

To determine

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

Answer to Problem 69E

The derivative of this function is obtained by: $4\pi r^2$ .

Explanation of Solution

Given:

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

Calculation:

A derivative is given by

  $f^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Therefore, the derivative of this function is obtained by:

  $\begin{array}{l}{V}^{\prime }\left(r\right)=\underset{h\to 0}{\lim }\frac{\frac{4}{3}\pi {\left(r+h\right)}^3-\frac{4}{3}\pi r^3}{h}\\ =\underset{h\to 0}{\lim }\frac{\frac{4}{3}\pi {\left(r^3+3r^{2}h+3h^{2}r+h^3\right)}^3-\frac{4}{3}\pi r^3}{h}\\ =\underset{h\to 0}{\lim }\frac{\frac{4}{3}\pi {\left(3r^{2}h+3h^{2}r+h^3\right)}^3}{h}\\ =\underset{h\to 0}{\lim }\frac{4}{3}\pi \left(3r^2+3hr+h^2\right)\\ V^{\prime }\left(r\right)=4\pi r^2\left(\text{By direct substitution}\right)\end{array}$

Conclusion:

Therefore,the derivative of this function is obtained by: $4\pi r^2$ .

(b)

To determine

To evaluate: the derivative when the radius is 4 inches.

Answer to Problem 69E

The derivative is $64\pi \approx 201.06$ .

Explanation of Solution

Calculation:

Using the equation for $V^{\prime }\left(r\right)$ found in part (a):

  $V^{\prime }\left(r\right)=4\pi r^2$

Now, substitute the value of radius

  $V\text{'}\left(4\right)=4\pi {\left(4\right)}^2=64\pi \approx 201.06$

Conclusion:

Therefore,the derivative is $64\pi \approx 201.06$ .

(c)

To determine

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

Answer to Problem 69E

  $V^{\prime }\left(r\right)$ would be measured in units of volume per units of distance

Explanation of Solution

Calculation:

The derivative of a function gives the slope of the tangent line to, or rate of change of, the function at a given point. So $V^{\prime }\left(r\right)$ would be measured in units of volume per units of distance (to measure radius).

Conclusion:

Therefore, $V^{\prime }\left(r\right)$ would be measured in units of volume per units of distance