# Problem of the Week

Problem E and Solution

A Rectangle and a Square

## Problem

Simeon has a rope that is \(108\) cm
long and is asked to cut the rope once so that one of the pieces can be
arranged, with its two ends touching, to form a square, and the other
piece can be arranged, with its two ends touching, to form a rectangle
with one side length of \(6\) cm.
Furthermore, the area of the square will be equal to the area of the
rectangle.

Where should Simeon make the cut to the original piece of rope?

## Solution

Let the length of the piece of rope used to form the square be \(4x\) cm. This is also equal to the
perimeter of the square. Then the side length of the square is \(4x\div 4=x\) cm. The area of the square is
\[x \times x=x^2\text{
cm}^2\tag{1}\]

The length of the piece of rope used to form the rectangle is \((108-4x)\) cm. This is also equal to the
perimeter of the rectangle. If one side length of the rectangle is 6 cm,
then there is \(108-4x-6-6=(96-4x)\) cm
left to form the lengths of the two other sides of the rectangle.
Therefore, the other side length of the rectangle is \(\frac{96-4x}{2}=(48-2x)\) cm. Thus, the
area of the rectangle is \[(6)(48-2x)=(288-12x)\text{
cm}^2\tag{2}\]

We are given that the area of the square is equal to the area of the
rectangle. So, by equating equations \((1)\) and \((2)\), we obtain \[\begin{aligned}
x^2&=288-12x\\
x^2+12x-228&=0\\
(x-12)(x+24)&=0
\end{aligned}\] Thus, \(x=12\)
or \(x=-24\). Since \(x\) is the length of the side of the
square, we must have \(x>0\).
Therefore, \(x=12\) cm. Then the length
of rope used to form the square is \(4x=4(12)=48\) cm.

Therefore, the cut should be made \(48\)
cm from one end (and so \(60\) cm from
the other end), creating a \(60\) cm
piece for the rectangle and a \(48\) cm
piece for the square.

Note:

The area of the square is \(12\times
12=144\text{ cm}^2\).

The length of the other side of the rectangle is \(48-2x=48-24=24\) cm. The area of the
rectangle is \(24\times 6=144\text{
cm}^2\).

(These calculations were not required but are provided as a check of the
correctness of the result.)