Chapter 2.2, Problem 76E

To determine

To draw: Possible designs for the rectangular garden with fencing and make a conjecture about the dimensions of the rectangular garden with the greatest possible area.

Answer to Problem 76E

The maximum area is 625 feet square.

Explanation of Solution

Given Information: There is 100 feet of fencing to enclose a rectangular garden.

Calculation:

The opposite sides of the rectangle are same. That is if one side is x, then the opposite side is also x. And if one of the remaining sides is y, the other side is also y.

Now, the perimeter of the rectangle is covered by the fencing 100. Therefore,

  $\begin{array}{c}2x+2y=100\\ x+y=50\\ y=50-x\end{array}$

The area of the rectangular garden is,

  $\begin{array}{c}\text{A}=\text{Length}\times \text{Breadth}\\ =x\times y\\ =x\left(50-x\right)\\ =50x-x^2\end{array}$

• Here, A represents the area of the rectangular garden.

Now, the maximum area is calculated by the y- coordinate of the vertex.

The x -coordinate of the vertex of the parabola $y=a{x}^2+bx+c$ is $x=-\frac{b}{2a}$ .

The x- coordinate of the parabola $A=50x-x^2$ is,

  $\begin{array}{c}x=-\frac{b}{2a}\\ =-\frac{50}{2\left(-1\right)}\left(a=-1,b=50\right)\\ =25\end{array}$

The maximum area is,

  $\begin{array}{c}A\left(25\right)=50\left(25\right)-{\left(25\right)}^2\\ =2{\left(25\right)}^2-{\left(25\right)}^2\\ ={\left(25\right)}^2\\ =625\end{array}$