Chapter 4, Problem 6CA

To determine

To calculate: The measure of the side of the given square.

Answer to Problem 6CA

The required equation is $f\left(x\right)=x^4-2x^3+x^2-18x-72$ .

Explanation of Solution

Given information:

The given diagram of the square and circle is:

  

The area of the shaded region is:

  $21.5\text{sq}\text{.}\text{m}$

Formula used:

Area of square:

  $A=sid{e}^2$

Area of circle:

  $A=\pi r^2$

Calculation:

Consider the area of the shaded region:

  $\begin{array}{c}{\left(2r\right)}^2-\pi r^2=21.5\\ 4r^2-\pi r^2=21.5\\ \left(4-\pi \right)r^2=21.5\\ r^2=\frac{21.5}{4-\pi }\end{array}$

Solve the above equation to find the value of r.

  $\begin{array}{c}{r}^2=\frac{21.5}{4-\pi }\\ r=5\end{array}$

Consider the side of the square from the diagram.

  $\begin{array}{c}side=2r\\ =2\times 5\\ =10\end{array}$

Hence, the side of the square is $10$ .