Answer
The length and width are 11 feet by 7 feet
Work Step by Step
Let us assume $ l $ to be the length and $ w $ to be the width of the rectangle such that:
$\begin{align}
& 2l+2w=36\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{I} \right) \\
& lw=77\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{II} \right)
\end{align}$
Divide equation (I) by $2$ and the value of $ l $ is,
$\begin{align}
& l+w=18 \\
& l=18-w\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{III} \right) \\
\end{align}$
Substitute the value of (I) in equation (II).
Then,
$\begin{align}
& \left( 18-w \right)w=77 \\
& 18w-{{w}^{2}}=77 \\
& {{w}^{2}}-18w+77=0
\end{align}$
Now, solve for $ w $.
Therefore,
$\begin{align}
& \left( {{w}^{2}}-11 \right)w-7w+77=0 \\
& \left( w-11 \right)w-7\left( w-11 \right)=0 \\
& w=11,7
\end{align}$
Substitute the value of $ w=11,7$ into equation (III):
If $ w=11$ then,
$\begin{align}
& l=18-11 \\
& =7
\end{align}$
If $ w=7$ then,
$\begin{align}
& l=18-7 \\
& =11
\end{align}$
Thus, the dimensions of the rectangle are 11 feet by 7 feet.
