suppose a wire two feet is to be cut into two pieces, with each piece to be formed into a square. Find the size of each piece to maximize the total area of the two squares.

Answer 1

Well suppose the two pieces are x, and (2-x) feet

Then the sides of the squares formed are (x/4) and (2-x)/4

and the total area

A = (x/4)² + ((2-x)/4)²

dA/dx = 1/16 [ 2x + 2 (2-x) (-1) ]

We need dA/dx = 0 for a minimum/maximum area giving 2x -2 (2-x) = 0

2x -4 + 2x = 0

4x-4 = 0

x = 1

So we cut the piece in half