A construction company wants to build a rectangular enclosure with an area of 1000 square feet by fencing in three sides and using its office building as the fourth side. Your objective as supervising engineer is to design the enclosure so that it uses the least amount of fencing. Proceed as follows. (a) Let x and y be the dimensions of the enclosure, where x is measured parallel to the building, and let L be the length of fencing required for those dimensions. Since the area must be 1000 square feet, we must have xy = 1000. Find a formula for L in terms of x and y, and then express L in terms of x alone by using the area equation. (b) Are there any restrictions on the value of x? Explain.

Answer 1

(a) L = x + 2y = x + 2000/x

(b) 0 < x < L, for obvious reasons

For optimum (minimum) L, dL/dx = 0

1 - 2000/x^2 = 0
x = sqrt2000 = 44.74 feet
y = 1000/x = 22.37 feet