i actually already have the answer, i'm just trying to figure out if this was an easy question or not, because i spent an hour+ on this question and couldn't figure it out lol

Area/perimeter formulas:
A_s = s*s
P_s = 4*s
A_c = pi*r^2
P_c = 2*pi*r

P_s + P_c = 50
4*s + 2*pi*r = 50
r = (50 - 4*s ) / (2*pi)

now we can replace one of the variables so that we get an equation with only 1 var instead of 2

A_c = pi*[ (50 - 4*s ) / (2*pi)]^2
P_c = 2 * pi * [ (50 - 4*s ) / (2*pi)]


and you want to minimize A_s + A_c

I don't feel like simplifying, but you can do that.

it looks like a parabola, so you can differentiate to find the critical points. or just use the algebra formula to find the minimum of a parabola in your domain s in (0, 50)

This post was edited by carteblanche on Dec 31 2013 06:31pm

for all regular polygons the more sides, the more efficient it is at enclosing area vs its perimeter. A circle could be described as an infinite sided polygon so if you wanted the max area for the given perimeter you should use 100% of the wire to make a circle. To minimize the area use 100% of the wire to make a square.

for the math behind the principle i stated, see carteblanche's post above. Also don't forget to check the end points of the domain when you're done!

you know, thinking about it, it might not be 100% square. So just calculate the places where the derivative is 0 and the end points and the answer will be obvious (see carteblanche's post)