Question

A running track of $440$ 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 lengths of its longest sides.

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

Answer 1

The correct option is B $110$

Let the length of the rectangle be x.
Let its breadth be y.
Then
Perimeter
$=2x+\pi \left(\frac{y}{2}\right)+\pi \left(\frac{y}{2}\right)$
$=2x+\pi .y$
$=440$
Or
$y=\frac{440-2x}{\pi }$
Now
Area of the rectangular part of the field
$A=x.y$
$=\frac{440x-2x^2}{\pi }$
Differentiating with respect to x, we get
$440-4x=0$
Or
$x=110$.ft.
Hence
The required length of the rectangular field $=110ft$.