Chapter 3: Q. 40 (page 288)

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.
An arena-style habitat whose front area is a semicircle and whose back area is rectangular, as shown previously at the right.

Expert verified

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

Step by step solution

given,

The maximum area is $2500$ square feet.

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

$\begin{aligned} P - 2 (x + y) - y + π (\frac{x}{2}) \\ P = 2 x + y + \frac{π x}{2} \end{aligned}$

Its area is,

$\begin{aligned} A = x y \\ 2500 = x y \\ y = \frac{2500}{x} \end{aligned}$

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

$\begin{aligned} P = 2 x + 2 \frac{2500}{x} - 20 \\ P = 2 x + \frac{5000}{x} - 20 \\ P^{'} (x) = 2 - \frac{5000}{x^{2}} \end{aligned}$

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

The length is,

$\begin{aligned} y = \frac{2500}{50} \\ y = 50 \end{aligned}$

The perimeter is,

$\begin{aligned} P = 2 (50) + 2 (50) - 20 \\ P = 100 + 100 - 20 \\ P = 180 \end{aligned}$

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

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