Chapter 3.1, Problem 72E

To determine

To find: The function that models the total area of the four pens and largest possible total area of the four pens.

Answer to Problem 72E

The function that models the total area of the four pens is equal to $g\left(x\right)=\frac{750x-5x^2}{2}$ and largest possible total area of the four pens is equal to $14062.5{\text{ft}}^2$ .

Explanation of Solution

Given information:

A rancher with $750\text{ft}$ of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of rectangle(see figure).

  

Calculation:

Assume that the breadth of the rectangular region is equal to $x\text{ft}$ and as the sides along the breadth are $5$ . So, the total fence for the five sides along breadth is equal to $5x\text{ft}$ .

The fence left is used to fence $2$ sides along the length of rectangular region. The left fence is equal to $750-5x\text{ft}$ . So, the length of the rectangular region is equal to $\frac{750-5x}{2}\text{ft}$ .

Use the formula for the area of the rectangular region.

  $\begin{array}{c}\text{A}=\text{L}\times \text{B}\\ \text{A}=\left(\frac{750-5x}{2}\right)\times x\\ \text{A}=\frac{750x-5x^2}{2}\end{array}$

So, the area function of the rectangular region is given by $g\left(x\right)=\frac{750x-5x^2}{2}$ . To maximize the area of the rectangular region graph the area function $g\left(x\right)=\frac{750x-5x^2}{2}$ .

To graph a function $y=\frac{750x-5x^2}{2}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{750x-5x^2}{2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{750}\boxed{\text{X}}\boxed{-}\boxed{5}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{)}\boxed{÷}\boxed{2}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(750\text{X}-5\text{X}^2\right)/2$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown in Figure (1).

  

Figure (1)

As observed from the graph, the area function has maximum value equal to $14062.5{\text{ft}}^2$ at $x=75$ .

Therefore, the function that models the total area of the four pens is equal to $g\left(x\right)=\frac{750x-5x^2}{2}$ and largest possible total area of the four pens is equal to $14062.5{\text{ft}}^2$ .