A rectangular field needs to be built outside of a gymnasium. Three walls of fencing are needed and the fourth wall is to be a wall of the gymnasium itself. The ideal area for such a field is exactly 360000ft^2. In order to minimize costs, it is necessary to construct the fencing using the least amount of material possible. Assuming that the material used in the fencing costs $78/ft, what is the least amount of money needednto build this fence of ideal area?
y = length of fencing, L = length, w = width,
minimize y
y = 2 w + L
so L = y - 2 w
A = w L = 360,000 ft^2
A = w (y-2w)
so
w y -2 w^2 = 360,000
w y = 2 w^2 + 360,000
y = 2 w + 360,000 / w
dy/w = 0 for min or max y
dy/dw = 0 = 2 - 360,000/w^2
2 w^2 = 360,000
w^2 = 180,000
w = 424.26 feet
find L
find 2w+L
multiply by the dollars/foot