Chapter 4.7, Problem 69PPSE

Solution

a.

To write an equation to find the dimensions of a square.

  $x^2-314=0$

Given:

Radius of circle $=\text{10 cm}$

Area of square = Area of circle

Formula Used:

Area of circle $=\pi r^2$

Area of square $=\text{side}\times \text{side}$

Calculation:

Finding the area of circle as:

  $\begin{array}{l}A=\pi r^2\\ \Rightarrow A=\frac{22}{7}\times {\left(10\right)}^2\\ \Rightarrow A=\frac{22}{7}\times 100\\ \Rightarrow A=3.14\times 100\\ \Rightarrow A=314{\text{ cm}}^{\text{2}}\end{array}$

Let the side of square $=x\text{ cm}$

Now, the area of square is:

  $\begin{array}{l}{A}_{{}_1}=x\times x\\ \Rightarrow A_{{}_1}=x^2\end{array}$

As these two areas are equal, the expression for $x$ is:

  $\begin{array}{l}{x}^2=314\\ \Rightarrow x^2-314=0\end{array}$

b.

To calculate the length of a side of the square.

The length of the side of a square is $17.72\text{ cm}\text{.}$

Given:

The equation is:

  $x^2-314=0$

Calculation:

Rewriting the equation as:

  $x^2=314$

Taking square root on both sides as:

  $\begin{array}{l}x=\pm \sqrt{314}\\ \Rightarrow x=\pm 17.72\end{array}$

The side of a square could be positive, so that the length of the side of a square is $17.72\text{ cm}\text{.}$