A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?
What do we know? The equation for the area of a rectangle is A = xy. We also know that the area will be 1,500,000 sq ft. We set up this equation:
xy = 1,500,000
Now, we need to model the length of fence needed. The perimeter is of the rectangle will be 2x + 2y. We also need an extra "side" of fence, so we will add an extra x feet to the total length of fence we need. (It can arbitrarily be x or y, but we choose x for simplicity).
Thus we have: L = 3x + 2y, where L is the length-of-fencing-we-will-need function. Let's minimize this. But first, let's get rid of the 'y' term and make L a function of strictly x. To do that, we take what we know, namely xy = 1,500,000 , and solve for y in terms of x. Doing this, we get . We plug this value in for y in L = 3x + 2y. We get: .
Now to minimize, we compute L' and find the critical points. Using the first-derivative test, you can figure out the correct answer. Good luck!