A wire of length 20 m is to be cut into two pieces. A piece of length ℓ 1 is bent to make a square of area A 1 and the other piece of length ℓ 2 is made into a circle of area A 2 . If 2 A 1 + 3 A 2 is minimum then π ℓ 1 : ℓ 2 is equal to:

Step-by-step solution

The length of wire is 20   m . According to the question: ℓ 1 + ℓ 2 = 20 ⇒ d ℓ 2   d ℓ 1 = - 1 Let the side of the square be x , then 4 x = ℓ 1 ⇒ x = ℓ 1 4       . . . 1 Let the radius of the circle be y , then 2 π y = ℓ 2 ⇒ y = ℓ 2 2 π       . . . 2 Hence, A 1 = ℓ 1 4 2 and A 2 = π ℓ 2 2 π 2 Let S = 2 A 1 + 3 A 2 ⇒ S = ( ℓ 1 ) 2 8 + 3 ( ℓ 2 ) 2 4 π ⇒