Question: A running track of 440 ft. is to be laid out enclosing a football field, the shape of which is a rec...

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 then the lengths of its sides are –

Answer 1

Solution

Perimeter = 440 ft.

2x + pr + pr = 440

or 2x + 2pr = 440 … (1)

A = Area of rectangular portion = x . 2r

A = x = $\frac { 1 } { \pi }$ (440x – 2x²)

$\frac { \mathrm { dA } } { \mathrm { dx } }$ = $\frac { 1 } { \pi }$ (440 – 4x) \ x = 110

= –ive

\ A is max. when x = 110

\ 2r = $\frac { 440 - 2 x } { \pi }$ = $\frac { 440 - 220 } { 22 / 7 }$ = 70.