Chapter 8.2, Problem 4P

To determine

To find:The factored form of the area of the shaded region.

Answer to Problem 4P

  $9x^2(4-\pi )$

Explanation of Solution

Given information: The side length of the square is $6x$ and the radius of the circle is $3x$ .

  

Calculation:

The area of the square is,

  $\begin{array}{c}{\text{A}}_1={\text{side}}^2\\ $ ={(6x)}^2$$ =36x^2$\end{array}$

The area of the circle is given by,

  $\begin{array}{c}{\text{A}}_2=\pi r^2\\ $ =\pi {(3x)}^2$$ =9\pi x^2$\end{array}$

Then, area of the shaded region is,

  $\begin{array}{l}{\text{A}}_1-{\text{A}}_2\\ =36x^2-9\pi x^2\\ =2\cdot 2\cdot 3\cdot 3\cdot x\cdot x-3\cdot 3\cdot \pi \cdot x\cdot x[\text{factoring each term]}\\ =3\cdot 3\cdot x\cdot x(2\cdot 2-\pi )[\text{factoring out the GCF]}\\ =9x^2(4-\pi )[\text{simplifying]}\end{array}$

Thus, factored form of the area of the shaded region is $9x^2(4-\pi )$ .