Having a rectangular parking lot with an area of 5,000 square yards that is to be fenced off on the three sides not adjacent to the highway, what is the least amount of fencing that will be needed to complete the job?

We can turn this into a constrained optimization problem. Let's denote x the side of the parking lot that is perpendicular to the highway, and y the side that is parallel to the highway. We therefore need to minimize f(x,y) = 2x + y, where xy = 5000.

From the second relation we see that we can represent y as being 5000 / x. So the problem becomes the unconstrained minimization of f(x) = 2x + 5000/x.

We differentiate f(x) with respect to x and we obtain 2 - 5000/x^2. Setting this to 0 yields 5000/x^2 = 2, so x^2 = 2500.

This equation gives 2 solutions x = 50, and x = -50, but we are only interested in the positive value, because the length of a fence cannot be negative. Knowing that y = 5000/x, we get our final solution: x = 50, y = 100.