you have 1000 yards of fencing and you plan to use the fencing to make 2 enclosures, one circular and one a square. How much of the 1000 yard of fencing should be used for each region if you want to maximize the combined area of both regions
Let x yards of fence be used for the circumference of the circle and the rest, (1000-x) yards, be used for the perimeter of the square
First of all, I think the question probably asks for the
minimum area, not the maximum area.
For the maximum area we would simply use all of the fence for
the circle, since the circle has the largest area for a given perimeter.
I would define the variables slightly differently.
Let r be the radius of the circle
let s be the side of the square
2πr + 4s = 1000
πr + 2s = 500
s = (500 - πr)/2
area = πr^2 + s^2
= πr^2 + (500 - πr)^2 /4
d(area)/dr = 2πr + (1/2)(500-πr)(-π)
= 0 for a max of area
2πr = -(1/2)(500-πr)(-π) , divide both sides by π, then times 2
4r = 500 - πr
4r + πr = 500
r(4+π) = 500
r = 500/(4+π) = appr. 70.012
so 2πr should be used for the circle, or 439.9
leaving 560.1 for the square to yield a minimum total area