A piece of wire 10 m long is to be cut into two pieces.

Question

A piece of wire 10 m long is to be cut into two pieces. One piece is to be bent

into a square and the other is to be bent into a circle. How should the wire be cut so

that the total area enclosed is a maximum.

Answer 1

A_total = A_sq + A_circle

Let x = length cut for circle

then 10-x = length available for the square

A_total = (x/(2π))²•π + ((x-10)/4)²

A_total = x²/(4π) + (1/16)(x²-20x+100)

dA_total / dx = 2x/(4π) + (1/16)(2x-20) (1)

d²A_total / dx² = 2/(4π) + (2/16) > 0 ===> A_total is concave UP, this means if we set dA / dx = 0 and solve, we are going to get a minimum value.

Check endpoints:

x=0 ==> A_total = (1/16)(100) ≈ 6.25 m²

x=10 ==> A_total = 10²/(4π) + (1/16)(10²-200+100) = 10² / (4π) ≈ 7.96 m²

To maximize area, all of the wire should be allocated for the circle.

If you meant to find the minimum area, set (1) to 0 and solve for x. I got x=4.399.