Chapter 9.8, Problem 9P

To determine

To find: the dimensions of the plot.

Answer to Problem 9P

The dimensions of the rectangular plot are 30 m by 45 m.

Explanation of Solution

Given:

Area $1350m^2$

  

Calculation:

The area of a rectangle is the product of its length and breadth. The length is given as y and breadth as x.

  $xy=1350$

The total length of fencing is the sum of the lengths of all the sides and given that it is $180m$ .

  $\begin{array}{l}y+x+y+x+x=180\\ 2y+3x=180\end{array}$

Divide both sides of the equation by 2.

  $\begin{array}{l}\frac{2y}{2}+\frac{3x}{2}=\frac{180}{2}\\ y+\frac{3x}{2}=90\end{array}$

Find the value of y from this equation.

  $y=90-\frac{3x}{2}$

Substitute the value of y in $xy=1350$ .

  $x\left(90-\frac{3}{2}x\right)=1350$

Simplify.

  $\begin{array}{l}90x-\frac{3}{2}{x}^2=1350\\ 180x-3x^2=2700\\ -3x^2+180x-2700=0\end{array}$

Divide both sides by $-3$ .

  $x^2-60x+900=0$

Factor the equation.

  ${\left(x-30\right)}^2=0$

Solve this equation.

  $x=30$

So, the breadth of the rectangle is 30 m.

Substitute 30 for x in $xy=1350$ and solve for y.

  $\begin{array}{l}30\left(y\right)=1350\\ y=\frac{1350}{30}\\ =45\end{array}$

The length of the rectangle is 45 m.

Therefore, the dimensions of the rectangular plot are 30 m by 45 m.

Conclusion:

The dimensions of the rectangular plot are 30 m by 45 m.