Chapter 3.4, Problem 4E

(a)

To determine

To calculate: The area of a square as a function of the radius of the circle.

Answer to Problem 4E

The area of the square is $A=2r^2$.

Explanation of Solution

Given Information: A square of side length s is inscribed in a circle of radius r.

Calculation:

The square is inscribed in a circle of radius r. Therefore, the diagonal of the square is equal to the diameter of the circle.

Each angle of a square is of ${90}^{\circ }$ . Since the diagonal bisects two angles, they are ${45}^{\circ }$ each in measurement. Therefore, angles of one half of the square are ${45}^{\circ },{45}^{\circ },\text{ and }{90}^{\circ }$.

Assume one side of the square be s.

  $\begin{array}{c}\sin 45=\frac{P}{H}\\ \frac{1}{\sqrt{2}}=\frac{\text{side of the square}}{\text{diagonal of the square}}\\ \frac{1}{\sqrt{2}}=\frac{s}{\text{Diameter of the circle}}\\ \frac{1}{\sqrt{2}}=\frac{s}{2r}\\ s=\sqrt{2}r\end{array}$

The area of the square is,

  $\begin{array}{c}\text{A}=s^2\\ ={\left(\sqrt{2}r\right)}^2\\ =2r^2\end{array}$ …… (1)

• Here, A is the area of the square and r is the radius of the circle.

Conclusion:

The area of the square is $A=2r^2$.

(b)

To determine

To calculate: The rate of change of area with respect to the radius r.

Answer to Problem 4E

The rate of change of area with respect to the radius r is $4r$.

Explanation of Solution

Given Information: The area of the square is $A=2r^2$.

Calculation:

The area of the square is,

  $A=2r^2$.

The rate of change of area is equal to the derivative of area with respect to the radius r ,

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

• Here, $\frac{dA}{dr}$ is the rate of change of area with respect to the radius r.

Conclusion:

The rate of change of area with respect to the radius r is $4r$.

(c)

To determine

To calculate: The rate of change of area with respect to r at $r=1\text{ and }r=8$

Answer to Problem 4E

The rate of change of area with respect to r at $r=1\text{ is }4$.

The rate of change of area with respect to r at $r=8\text{ is }32$.

Explanation of Solution

Given Information: The rate of change of area with respect to the radius is,

  $\frac{dA}{dr}=4r$

Calculation:

: The rate of change of area with respect to the radius is,

  $\frac{dA}{dr}=4r$

The rate of change of area with respect to r at $r=1$ is,

  $\begin{array}{c}{\left[\frac{dA}{dr}\right]}_{r=1}={\left[4r\right]}_{r=1}\\ =4\end{array}$

• Here, ${\left[\frac{dA}{dr}\right]}_{r=1}$ is the rate of change of area with respect to r when r is 4.

The rate of change of area with respect to r at $r=8$ is,

  $\begin{array}{c}{\left[\frac{dA}{dr}\right]}_{r=8}={\left[4r\right]}_{r=8}\\ =32\end{array}$

• Here, ${\left[\frac{dA}{dr}\right]}_{r=8}$ is the rate of change of area with respect to r when r is 8.

Conclusion:

The rate of change of area with respect to r at $r=1\text{ is }4$.

The rate of change of area with respect to r at $r=8\text{ is }32$.

(d)

To determine

To calculate: the appropriate units for $\frac{dA}{dr}$.

Answer to Problem 4E

The appropriate unit for $\frac{dA}{dr}=\text{in}$.

Explanation of Solution

Given Information: The rate of change of area with respect to the side.

Calculation: r is measured in inches and A is measured in square inches.

The rate of change of area with respect to the area is,

  $\frac{dA}{dr}$

Substitute the units in above equation,

  $\begin{array}{c}\frac{dA}{dr}=\frac{{\text{in}}^2}{\text{in}}\\ =\text{in}\end{array}$

Conclusion:

The appropriate unit for $\frac{dA}{dr}=\text{in}$.