Chapter 2.4, Problem 29E

To determine

To find: The rate of change of area of circle with respect to radius when $r=3\text{in}\text{.}$.

Answer to Problem 29E

The rate of change of area of circle with respect to radius when $r=3\text{in}\text{.}$ is equal to $6\pi \text{in}{\text{.}}^{\text{2}}/\text{in}\text{.}$.

Explanation of Solution

Given information:

The radius of the circle is $3\text{in}\text{.}$.

Calculation:

The formula for the area of the circle is equal to $\pi r^2$.

The rate of change of area at $r=3\text{in}\text{.}$ is the instantaneous rate of change of area function at $r=3\text{in}\text{.}$.

The formula for the instantaneous rate of change of area with respect to radius at $r\text{in}\text{.}$ is equal to $\underset{h\to 0}{\lim }\frac{f\left(r+h\right)-f\left(r\right)}{h}$.

The formula for the instantaneous rate of change of $y=\pi r^2\text{in}\text{.}$ at $r=3\text{in}\text{.}$ is equal to $\underset{h\to 0}{\lim }\frac{f\left(3+h\right)-f\left(3\right)}{h}$.

Substitute $3+h$ for $r$ in the function.

  $\begin{array}{c}f\left(r\right)=\pi r^2\\ f\left(3+h\right)=\pi {\left(3+h\right)}^2\\ f\left(3+h\right)=\pi \left(9+h^2+6h\right)\\ f\left(3+h\right)=\pi h^2+6\pi h+9\pi \end{array}$

Substitute $3$ for $r$ in the function.

  $\begin{array}{c}f\left(r\right)=\pi r^2\\ f\left(3\right)=\pi {\left(3\right)}^2\\ f\left(3\right)=\pi \left(9\right)\\ f\left(3\right)=9\pi \end{array}$

Substitute $\pi h^2+6\pi h+9\pi$ for $f\left(3+h\right)$ and $9\pi$ for $f\left(3\right)$ in the formula.

  $\begin{array}{c}\underset{h\to 0}{\lim }\frac{f\left(3+h\right)-f\left(3\right)}{h}=\underset{h\to 0}{\lim }\frac{\pi h^2+6\pi h+9\pi -9\pi }{h}\\ =\underset{h\to 0}{\lim }\frac{\pi h^2+6\pi h}{h}\\ =\underset{h\to 0}{\lim }\frac{h\left(\pi h+6\pi \right)}{h}\\ =\underset{h\to 0}{\lim }\left(\pi h+6\pi \right)\end{array}$

Further simplify.

  $\begin{array}{c}\underset{h\to 0}{\lim }\frac{f\left(3+h\right)-f\left(3\right)}{h}=\pi \left(0\right)+6\pi \\ =6\pi \end{array}$

Therefore, the rate of change of area of circle with respect to radius when $r=3\text{in}\text{.}$ is equal to $6\pi \text{in}{\text{.}}^{\text{2}}/\text{in}\text{.}$.