Chapter 14.7, Problem 16HP

To determine

To calculate: Theexpression in factored form of the area of the shaded portion.

Answer to Problem 16HP

The area of the shaded region is $9x^2\left(25\pi -9\right)\text{square}\text{unit}$ .

Explanation of Solution

Given Information:

  $\text{Radius}\text{of}\text{the}\text{circle}\left(\text{r}\right)=15x$

Width of the walkway length of sides of the square $=9x$

Formula Used:

  $\text{Area}\text{of}\text{a}\text{circle}\left(A\right)=\pi r^2$ .

  $\text{Area}\text{of}\text{a}\text{square}=l^2,l$ is the length of sides of the square

Calculation:

The given sides of the squared base is $9x$ .

Find the area. Use the formula of the area of the square.

  $\begin{array}{c}\text{Area}\text{of}\text{the}\text{square}=l^2\\ ={\left(9x\right)}^2\\ =81x^2\text{square}\text{unit}\end{array}$

The radius of the circle is $15x$ .

Now, find the area. Use the formula of the area of the circle.

  $\begin{array}{c}\text{Area}\text{of}\text{the}\text{circle}=\pi {\left(15x\right)}^2\\ =225\pi x^2\text{square}\text{unit}\end{array}$

Find the area of the shaded region, subtract area of the square from the area of the circle.

  $\begin{array}{c}\text{Area}\text{of}\text{the}\text{shaded}\text{region}=\text{Area}\text{of}\text{the}\text{circle}-\text{Area}\text{of}\text{the}\text{square}\\ \text{=225}\pi x^2-81x^2\\ =9x^2\times 25\pi -9x^2\times 9\\ =9x^2\left(25\pi -9\right)\text{square}\text{unit}\end{array}$

Hence, the area of the shaded region is $9x^2\left(25\pi -9\right)\text{square}\text{unit}$ .