Question

A 2 feet piece of wire is cut into two pieces and once piece is bent into a square and the other is bent into an equilateral triangle. How much wire should be used for the square to ensure that the total area enclosed by both shapes is minimized?

Answer 1

if the triangle has side a, and the square has side b,

3a+4b=2
The total area is
A = (1/2)(a/2)(a/2 √3) + b^2
16A = 2√3 a^2 + (4b)^2
= 2√3 a^2 + (2-3a)^2
16 dA/da = 4√3 a - 6(2-3a)

dA/da = 0 when a = 6/(9+2√3)