Answer
The length and width of the rug are 12 feet by 9 feet
Work Step by Step
Let us assume $ l $ be the length and $ w $ be the width of the rectangle such that:
$\begin{align}
& lw=108\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{I} \right) \\
& {{l}^{2}}+{{w}^{2}}={{15}^{2}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{II} \right)
\end{align}$
Solving equation (I):
$ l=\frac{108}{w}$ (III)
Substitute the value of $ l $ in equation (II).
Then,
$\begin{align}
& \frac{{{108}^{2}}}{{{w}^{2}}}+{{w}^{2}}=225 \\
& 11664+{{w}^{4}}-225{{w}^{2}}=0 \\
& \left( {{w}^{2}}-81 \right)\left( {{w}^{2}}-144 \right)=0
\end{align}$
Now, $\left( {{w}^{2}}-81 \right)=0\ \text{ or }\left( {{w}^{2}}-144 \right)=0$. All the possible values of $ w $ are $ w=\pm 9,\pm 12$. Negative value are not possible.
Therefore,
$ w=9,12$
If $ w=9$ then,
$\begin{align}
& l=\frac{108}{9} \\
& =12
\end{align}$
If $ w=12$ then,
$\begin{align}
& l=\frac{108}{12} \\
& =9
\end{align}$
Thus, the dimensions are 12 feet by 9 feet.
