Question

A running track of $440\mathrm{ft}$. is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end.

If the area of the rectangular portion is to be maximum find the length of its longest sides.

• A. $55$
• B. $110$
• C. $220$
• D. $140$

Answer 1

The correct option is B

$110$

Step 1. Explanation for the correct option.

Let the length and breadth of the rectangle be $x$ and $y$.

Perimeter of semicircle$=\mathrm{\pi }\times \frac{\mathrm{y}}{2}$

Total perimeter

$$\begin{gathered} P=2x+\frac{\mathrm{\pi y}}{2}+\frac{\mathrm{\pi y}}{2} \\ \Rightarrow 440=2x+\mathrm{\pi y} \\ \therefore y=\frac{440-2x}{\mathrm{\pi }} \end{gathered}$$

Area of rectangle:

$$\begin{gathered} A=xy \\ =x\left(\frac{440-2x}{\mathrm{\pi }}\right) \\ =\frac{440x-2x^2}{\mathrm{\pi }} \end{gathered}$$

Differentiate w.r.t. $x$:

$\frac{dA}{dx}=\frac{440-4x}{\mathrm{\pi }}$

For maximum/minimum value $\frac{dA}{dx}=0$

$$\begin{gathered} \frac{440-4x}{\mathrm{\pi }}=0 \\ \Rightarrow 440-4x=0 \\ \Rightarrow 440=4x \\ \therefore x=110 \end{gathered}$$

And $\frac{d^{2}A}{d{x}^2}=\frac{-4}{\mathrm{\pi }}<0$

Thus, the length of its longest sides is $110$.

Step 2. Explanation for the incorrect options.

As the length of its longest sides is $110$ as determined above.

So, option A, option C, and option D are an incorrect option.

Hence, the correct option is $\mathrm{B}$.