Answer
The dimensions of the rectangular plot are $ x=12\text{ feet, }y=15\text{ feet}$ or
$ x=\frac{15}{2}\text{ feet, }y=24\text{ feet}$.
Work Step by Step
The length of the plot is y and the width is x.
Hence, the perimeter of three sides is the sum of all the sides of the rectangular plot which is as shown below:
$2x+y $ and it is given as $39$ feet, that is,
$2x+y=39$ (I)
So, the area of the rectangular plot is 180 square feet, which is, $ length\times breadth $
$ xy=180$ (II)
From equation (II), obtain the value of y:
$ y=\frac{180}{x}$
Put the value of y in equation (I) to obtain the value of x:
$2x+\frac{180}{x}=39$
Simplify
$2{{x}^{2}}-39x+180=0$
Factorize the equation and solve as shown below:
$\begin{align}
& 2{{x}^{2}}-24x-15x+180=0 \\
& 2x\left( x-12 \right)-15\left( x-12 \right)=0 \\
& \left( x-12 \right)\left( 2x-15 \right)=0
\end{align}$
Equate each factor to 0, which gives
$\left( x-12 \right)=0$ or $\left( 2x-15 \right)=0$
Therefore, $ x=12$ or $ x=\frac{15}{2}$
Put $ x=12$ in equation (II); this gives
$ y=15$
Substitute $ x=\frac{15}{2}$ in equation (II); this gives
$ y=24$
Therefore, the dimensions of the rectangular plot are either $ x=12\text{ feet, }y=15\text{ feet}$ or $ x=\frac{15}{2}\text{ feet, }y=24\text{ feet}$.
