use all of the wire

Question

You are given 200 cm of wire, and you must construct two items: a square-shaped frame for a window, and a circular TV antenna. Because you must ship these items it is important to minimize the total area of the objects. How much wire should be used for each object to accomplish this? You must use all of the wire.

Answer 1

R+s = 200

Area = S^2 + pi * R^2

= S^2 + pi * (200-s)^2

= S^2 + pi ( 40000 - 400s + s^2)

= s^2 + pi * s^2 - 400*pi * s + 40000*pi

= (1+pi) s^2 - 400*pi * s + 40000*pi

= A(s)

0 = dA/ds = 2 ( 1 + pi) s - 400*pi

400*pi = 2(1+pi)*s

400*pi / [2(1+pi)] = s

S = 151.70939859895522906888613780898

R = 48.290601401044770931113862191021