Q. 39 - Calculus

Question

You are in charge of constructing a zoo habitat for prairie dogs, with the requirement that the habitat must enclose $2500$ square feet of area and use as little border fencing as possible. For each of the habitat designs described below, find the amount of border fencing that the project will require. 

A rectangular habitat with a $20$-foot-wide nesting hutch along one side (so that fencing is not needed along those $20$ feet), as shown next at the left. 

Step-by-Step Solution

Answer

Ans:  The amount of fencing required is $180$ feet.

Step 1. Given information.

given,  

        The maximum area is $2500$ square feet.
        The width of the nesting hutch is $20$ feet

Step 2. The objective is to find the amount of border fencing required excluding the nesting hutch.

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

    $\begin{array}{r}P=2(x+y)-20\\ P=2x+2y-20\end{array}$

Its area is,

   $\begin{array}{r}A=xy\\ 2500=xy\\ y=\frac{2500}{x}\end{array}$

Putting $y=\frac{2500}{x}\text{ in }P=2x+2y-20$

    $\begin{array}{r}P=2x+2\frac{2500}{x}-20\\ P=2x+\frac{5000}{x}-20\\ P^{\mathrm{\prime }}\left(x\right)=2-\frac{5000}{x^2}\end{array}$

Step 3. Equating 2 − 5000 x 2  to  0 ,

$\begin{array}{c}2-\frac{5000}{x^2}=0\\ -\frac{5000}{x^2}=-2\\ \frac{5000}{x^2}=2\\ 2x^2=5000\\ x=50\end{array}$

Step 4. Finding length and perimeter.

The length is,

      $\begin{array}{r}y=\frac{2500}{50}\\ y=50\end{array}$

The perimeter is,

     $\begin{array}{r}P=2\left(50\right)+2\left(50\right)-20\\ P=100+100-20\\ P=180\end{array}$

Therefore, the amount of fencing required is $180$ feet.