1. ## optimization problem

A piece of wire 10m long is cut into two pieces. One is bent into a square and the other into a circle. How should the wire be cut so that the area enclosed is (a) a maximum? (b) minimum?

I need help on setting up the original equation. I know how to find derivatives, maximums and minimums.

2. Originally Posted by cityismine
A piece of wire 10m long is cut into two pieces. One is bent into a square and the other into a circle. How should the wire be cut so that the area enclosed is (a) a maximum? (b) minimum?

I need help on setting up the original equation. I know how to find derivatives, maximums and minimums.
Let x = length of one piece, in meters.
So, (10 -x) = lentgh of the other piece, in meters.

say the x-piece be bent into a squre. Then the (10 -x) piece be bent into a circle.

For the square:
Perimeter = 4s = x
So, s = x/4
And so, area enclosed, A1 = s^2 = (x/4)^2 = (1/16)x^2

For the circle:
Perimeter = circumference = (2pi)r = 10 -x
So, r = (10 -x)/(2pi)
And so, area enclosed, A2 = (pi)r^2 = (pi)[(10 -x)/(2pi)]^2
A2 = (1/(4pi))[10 -x)^2]

Hence, total enclosed area,
A = A1 +A2
A = (1/16)x^2 +(1/(4pi))[(10 -x)^2] ------**

You now take over.