Question

A running track of length $440$ m is to be laid around 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, then the lengths of its sides is

• A. $\frac{220}{\pi },110$
• B. $\frac{220}{\pi },220$
• C. $\frac{110}{\pi },220$
• D. $\frac{110}{\pi },110$

Answer 1

The correct option is A $\frac{220}{\pi },110$

Area of rectangular portion $=2rx$
Perimeter of track $=440$ m
$\therefore 440=2x+2\pi r\cdots \left(i\right)$
So, Area of rectangular portion $=2r(220-\pi r)$
$\therefore f\left(r\right)=440r-2\pi r^2$
Area to be maximum $\Rightarrow f^{\prime }\left(r\right)=0$
$\Rightarrow 440-4\pi r=0$
$\Rightarrow r=\frac{110}{\pi }$
$f^{\prime \prime }\left(r\right)=-4\pi <0$
$\Rightarrow x=110$
$\therefore$ sides of rectangle are $2r=\frac{220}{\pi }$ m and $x=110$ m