Question

A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.

Answer 1

Let 2a and 2b be the sides of rectangular portion ABCD.

Given length of running track = 440 m

πa + πb + πa + πb = 440

`\implies` 2πa + 2πb = 440

`\implies 2 xx 22/7(a + b)` = 440

`\implies` a + b = 70    ...(1)

Area of Rectangle ABCD

A = (2a)(2b)

= 4ab

= 4a(70 – a)

= 4(70a – a²)

∴ `("dA")/("da")` = 4(70 – 2a)

For max/min `("dA")/("da")` = 0

4(70 – 2a) = 0

∴ a = 35

Also `("d"^2"A")/("da"^2)` = – 8a

= – 8 × 35 < 0

So, A is maximum when a = 35

By (1),

b = 70 – a

= 70 – 35

= 35

Hence sides of rectangular portion are

2a = 2 × 35 = 70 m

and 2b = 2 × 35 = 70 m

2^nd part: Area of whole field

= `2 xx 1/2 π"a"^2 + 2 xx 1/2 π"b"^2 + 2"a" xx 2"b"`

= πa² + πb² + 4ab

= `22/7 xx (35)^2 + 22/7 xx (35)^2 + 4 xx 35 xx 35`

= 3850 + 3850 + 4900

= 12600