Q. 35 - Calculus

Question

A farmer wants to build four fenced enclosures on his farmland for his free-range ostriches. To keep costs down, he is always interested in enclosing as much area as possible with a given amount of fence. For the fencing projects given below,  determine how to set up each ostrich pen so that the maximum possible area is enclosed, and find this maximum area.   

          A rectangular ostrich pen built with $350$ feet of fencing material. 

Step-by-Step Solution

Answer

Ans:  The maximum area is, $7656.25\text{ square feet }$

Step 1. Given information.

given,  

        The perimeter of the rectangular ostrich pen is $350$ feet.

Step 2. The objective is to determine the maximum possible area enclosed by the rectangular ostrich pen.

Let, the width be $x$ feet and the length be $y$ feet.

So its perimeter is,

         $\begin{array}{r}2(x+y)=350\\ x+y=175\end{array}$

Solving the equation for $y$,

           $\begin{array}{r}x+y=175\\ y=175-x\end{array}$

Now, its area is 

$\begin{array}{r}A=xy\\ =x(175-x)\\ =175x-x^2\end{array}$

Step 3. To maximize the area find it's derivative first.

So,

         $\begin{array}{r}A=\frac{d}{dx}\left(175x-x^2\right)\\ A=-2x+175\end{array}$$\begin{array}{r}{A}^{\mathrm{\prime }}=\frac{d}{dx}\left(175x-x^2\right)\\ A^{\mathrm{\prime }}=-2x+175\end{array}$

Equating to $0$

       $\begin{array}{r}-2x+175=0\\ 2x=175\\ x=\frac{175}{2}\end{array}$

Step 4. Putting x = 175 2  in  y = 175 − x

$\begin{array}{r}y=175-\frac{175}{2}\\ =\frac{175}{2}\end{array}$

So, the length and width come to be equal.

Hence, a square pen with sides $\raisebox{1ex}{$175$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ feet will produce the maximum area.

The maximum area is,

           $\begin{array}{r}A=\frac{175}{2}\times \frac{175}{2}\\ A=7656.25\end{array}$

Therefore, the maximum area is $7656.25$ square feet.