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

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 side are

TestHub · Accepted Answer

Perimeter = 440 ft ⇒ 2 x + π r + π r = 440 2 x + 2 π r = 440 A A ​ = Area of the rectangular portion = x 2 r = x π ( 440 − 2 x ) ​ = π 1 ​ ( 440 x − 2 x 2 ) ​ Let d x d A ​ = π 1 ​ ( 440 − 4 x ) = 0 ​ ⇒ x = 110 for which d x 2 d 2 A ​ < 0 ⇒ A is maximum when x = 110 ⇒ 2 r = π 440 − 2 x ​ = 22/7 440 − 220 ​ = 70 ∴ r = 35 ft and x = 110 ft ​

Mathematics - Application of Derivative Question with Solution | TestHub

Options:

Doubts & Discussion