Chapter 4.1, Problem 48PPSE

Solution

To state: A quadratic function to represent the area of all rectangles with a perimeter of 36 ft. Then graph the function and describe the rectangle that has the largest area.

The quadratic function that represents the area of all rectangles is $A=x\left(18-x\right)$ and the maximum area is 81 square feet.

Given information:

The given perimeter of a rectangle is 36ft.

Explanation:

Let’s assume x as the length of the rectangle and y as the width of the rectangle.

Then the perimeter of the rectangle is:

  $\begin{array}{c}2\left(x+y\right)=36\\ x+y=\frac{36}{2}\\ x+y=18\\ y=18-x\end{array}$

Now the area of the rectangle is:

  $\begin{array}{l}A=xy\\ A=x\left(18-x\right)\end{array}$

Now plot the function $A=x\left(18-x\right)$:

This function will have value 0 at,

  $\begin{array}{c}x\left(18-x\right)=0\\ x=0\text{or}18-x=0\\ x=0\text{or}x=18\end{array}$

  

The resultant graph looks like a parabola and it has its maximum at $x=9$ .

Then,

  $\begin{array}{c}x+y=18\\ 9+y=18\\ y=9\end{array}$

Now the maximum area will be:

  $\begin{array}{l}A=xy\\ A=9\cdot 9\\ A=81{\text{ft}}^2\end{array}$

Therefore, the maximum area is 81 square feet.