Question

The width of a rectangular fence is $5\mathrm{feet}$ less than the length. If the length is decreased by $3\mathrm{feet}$ and the width is increased by $1\mathrm{feet}$ , the area limited by the new fence will be the same as the area of the original fence. Find the dimensions of the original rectangular fence.

Answer 1

Step-1: Find the area of the original rectangular fence:

The width of a rectangular fence is $5\mathrm{feet}$ less than the length.

If the length is decreased by $3\mathrm{feet}$ and the width is increased by $1\mathrm{feet}$ , the area limited by the new fence will be the same as the area of the original fence.

Let $x\mathrm{feet}$ be the length of the original rectangular fence.

Since the width of a rectangular fence is $5\mathrm{feet}$ less than the length.

It follows that, the width of the original rectangular fence will be $\left(x-5\right)\mathrm{feet}$.

The formula for the area of the original rectangular fence is,

$\mathrm{Area}=\mathrm{length}\times \mathrm{width}$

Substitute the values of length and width in the area.

$\begin{array}{rcl}\mathrm{Area}& =& \left(x\right)\times \left(x-5\right)\\ & \Rightarrow & \mathrm{Area}=x\left(x-5\right)\\ & \Rightarrow & \mathrm{Area}=\left(x^2-5x\right){\mathrm{ft}}^2......\left(1\right)\end{array}$

Step-2: Find the area of the new fence:

If the length is decreased by $3\mathrm{feet}$ and the width is increased by $1\mathrm{feet}$ .

It follows that, the length of the new fence will be $\left(x-3\right)\mathrm{feet}$.

And the width of the new fence will be $\left[\left(x-5\right)+1\right]\mathrm{feet}$.

The formula for the area of the new fence is,

$\begin{array}{rcl}\mathrm{Area}& =& \mathrm{length}\times \mathrm{width}\\ & \Rightarrow & \mathrm{Area}=\left(\mathrm{x}-3\right)\times \left[\left(\mathrm{x}-5\right)+1\right]\\ & \Rightarrow & \mathrm{Area}=\left(\mathrm{x}-3\right)\left[\mathrm{x}-5+1\right]\\ & \Rightarrow & \mathrm{Area}=\left(\mathrm{x}-3\right)\left(\mathrm{x}-4\right)\\ & \Rightarrow & \mathrm{Area}={\mathrm{x}}^2-4\mathrm{x}-3\mathrm{x}+12\\ & \Rightarrow & \mathrm{Area}=\left({\mathrm{x}}^2-7\mathrm{x}+12\right){\mathrm{ft}}^2......\left(2\right)\end{array}$

Step-3: Find the dimensions of the original rectangular fence:

Since the area limited by the new fence will be the same as the area of the original fence.

It follows that, $\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{original}\mathrm{fence}=\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{new}\mathrm{fence}$.

$\begin{array}{rcl}{x}^2-5x& =& x^2-7x+12\\ & \Rightarrow & x^2-5x-x^2+7x=12\\ & \Rightarrow & 2x=12\\ & \Rightarrow & x=\frac{12}{2}\\ & \Rightarrow & x=6\end{array}$

Substitute $x=6$ in the width of the rectangular fence.

$\begin{array}{rcl}x-5& =& 6-5\\ & =& 1\end{array}$

Therefore, the length and width of the original rectangular fence will be $\mathbf{6}\mathbf{}\mathbf{feet}$ and $\mathbf{1}\mathbf{}\mathbf{feet}$ respectively.