Chapter 3.1, Problem 71E

To determine

To find: The dimensions of the field of largest area that a farmer can fence.

Answer to Problem 71E

The dimensions of the field of largest area that the farmer can fence is $600\text{ft}$ by $1200\text{ft}$ .

Explanation of Solution

Given information:

A farmer has $2400\text{ft}$ of fencing and wants to fence off a rectangular field that borders a straight river. The farmer doesn’t fence along the river.

Calculation:

As per given, assume that the length of the rectangular field is equal to $x\text{ft}$ .

As the length is fenced two times i.e. length of the fence for these two sides is equal to $2x\text{ft}$ and as the total fence is $2400\text{ft}$ . So, the breadth of the rectangular field is equal to $2400-2x\text{ft}$ .

Use the formula for the area of the rectangular region.

  $\begin{array}{l}\text{A}=\text{L}\times \text{B}\\ \text{A}=x\times \left(2400-2x\right)\\ \text{A}=2400x-2x^2\end{array}$

So, the area function of the rectangular region is given by $g\left(x\right)=2400x-x^2$ . To maximize the area of the rectangular region graph the area function $g\left(x\right)=2400x-x^2$ .

To graph a function $y=2400x-2x^2$ , 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 $y=2400x-2x^2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2400}\boxed{\text{X}}\boxed{-}\boxed{2}\boxed{\text{X}}\boxed{^}\boxed{2}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=2400\text{X}-2\text{X}^2$

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 at $x=600$ .

So, the length of the rectangular region is equal to $600\text{ft}$ and the breadth of the rectangular region is equal to $2400-2\left(600\right)=1200\text{ft}$ .

Therefore, the dimensions of the field of largest area that the farmer can fence is $600\text{ft}$ by $1200\text{ft}$ .