posted by mark .
A piece of wire 24 cm long is cut into two pieces and each piece is bent to form a square . Find the lenght of each piece of wire in order to maximize the sum of the area of the two squares?
if the squares have side x and y,
y = 6-x
area = x^2+y^2 = x^2 + (6-x)^2 = 2x^2 - 12x + 36
Now, that's a parabola which opens upward, so it has no maximum value. Since 0<=x<=6, the maximum area is reached at the endpoints of the interval.
So, maximum area is when there is just one large square, area 36.
If you must have two nonzero pieces of wire, then choose one to be very tiny and the other to be almost 24 cm.