We have a piece of wire that is 50cm long and we're going to cut it into two pieces. One piece will be bent into a square and the other will be bent into a circle. Determine where the wire should be cut so that the enclosed areas will be a maximum. Note that it is possible to have the whole piece of wire go the square or to the circle (meaning not cut it at all)

Question

anonymous · Answer

Area of a square = ?
Area of a circle = ?

anonymous · Answer

Don't cut the wire

anonymous · Answer

Its an optimization problem. I know you have you incorporate the area of both. I just don't know how to begin the problem.

anonymous · Answer

to*

anonymous · Answer

Well, the length of the wire can be split into two pieces. One for square and one for circle. The length of first piece piece represents perimeter of square. The length of remaining piece (50 - whatever first piece) represents circumference of circle.

Then, you have area of square and area of circle add up. Now, you need to maximize. Which means, you set the differential to zero.

anonymous · Answer

Let us call S as allocated to square.
S = 4L where L is length of each side of square.
So, L = S/4.

Then, 50-S is allocated to circle.
2pi*r = 50-S
=> r = (50-S)/2pi

Total area = L^2 + pi * r^2
=> (S/4)^2 + pi * (50-S/2pi)^2

Now differentiate and equate to zero.

anonymous · Answer

Okay, thats what i thought for the first step. I just didnt know how to set the equation to maximize. Let me see if it works. thanks