Answer
The length is 18 inches and the width is 12 inches.
Work Step by Step
We start with:
$\begin{align}
& lw=216\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{I} \right) \\
& 2\left( l-4 \right)\left( w-4 \right)=224\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{II} \right)
\end{align}$
Solving equation (I)
Then,
$ l=\frac{216}{w}$ (III)
Substitute the value of l in equation (II):
$\begin{align}
& \frac{216}{w}w-4\frac{216}{w}-4w-96=0 \\
& 216+\frac{864}{w}-4w-96=0 \\
& 4{{w}^{2}}+864-120w=0 \\
& {{w}^{2}}-30w+216=0
\end{align}$
Now, solve:
$\begin{align}
& \left( w-12 \right)\left( w-18 \right)=0 \\
& w=12,\ 18
\end{align}$
Substitute the value of $ w=12,18$ into equation (III).
Then,
$\begin{align}
& l=\frac{216}{12} \\
& =18
\end{align}$
And
$\begin{align}
& l=\frac{216}{18} \\
& =12
\end{align}$
Hence, the length is 18 inches and the width is 12 inches.
