College Algebra, Chapter 1, 1.3, Section 1.3, Problem 100

Suppose that a wire $360 in$ long is cut into two pieces. One piece is formed into a square and the other is formed into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire?

If we let $x$ be the length of the wire we take to form square then $360 - x$ will be the length of the wire we take to form the circle.

Recall that the perimeter of the square is $4s$ where $s$ is the side of the square, and the circumference of the circle is equal to $2 \pi r$ where $r$ is the radius.

Also, the area of the square given the length $s$ is $A = s^2$, while the area of the circle with radius $r$ is $A = \pi r^2$. If the two areas are equal, then

$s^2 = \pi r^2$

We know that divided lengths represent the perimeter of each figure.

$
\begin{array}{ccc}
\text{For circle,} & \text{For square,} & \\
360 - x = 2 \pi r & x = 4s & \text{Divide both sides by } 4 \\
& \frac{x}{4} = s & \text{Square both sides} \\
& \left( \frac{x}{4} \right)^2 = s^2 & \text{Substitute } s^2 = \pi r^2 \\
& \left( \frac{x}{4} \right)^2 = \pi r^2 & \text{Solve for } r \\
& r = \sqrt{\frac{\displaystyle \left( \frac{x}{4} \right)^2 }{\pi}} & \\
& r = \frac{x}{4 \sqrt{\pi}} &
\end{array}
$

By using the perimeters of the circle,


$
\begin{equation}
\begin{aligned}

360 - x =& 2 \pi r
&&
\\
\\
360 - x =& 2 \pi \left( \frac{x}{4 \sqrt{r}} \right)
&& \text{Substitute } r = \frac{x}{4 \sqrt{r}}
\\
\\
360 - x =& \frac{\pi x}{2 \sqrt{\pi}}
&& \text{Apply cross multiplication}
\\
\\
720 \sqrt{\pi} - 2 \sqrt{\pi} x =& \pi x
&& \text{Combine like terms}
\\
\\
720 \sqrt{\pi} =& \pi x + 2 \sqrt{\pi} x
&& \text{Factor out } x
\\
\\
720 \sqrt{\pi} =& x (\pi + 2 \sqrt{\pi})
&& \text{Solve for } x
\\
\\
x =& \frac{720 \sqrt{\pi}}{\pi + 2 \sqrt{\pi}} in
&&

\end{aligned}
\end{equation}
$


Therefore, the length of the wire for the square is $\displaystyle x = \frac{720 \sqrt{\pi}}{\pi + 2 \sqrt{\pi}} in$ while the length of the wire for circle is $360 - x = 360 - \displaystyle \frac{720 \sqrt{\pi}}{\pi + 2 \sqrt{\pi}} = \frac{360 \pi + 720 \sqrt{\pi} - 720 \sqrt{\pi}}{\pi + 2 \sqrt{\pi}} = \frac{360 \pi}{\pi + 2 \sqrt{\pi}} in$