What is the maximum possible area of the pasture?

Question

The back of Alisha's property is a creek. Alisha would like to enclose a rectangular area, using the creek as one side and fencing for the other three sides, to create a pasture. If there is 380 feet of fencing available, what is the maximum possible area of the pasture?

Answer 1

This is a classic problem of the type of min/max problem. In this case there is a slight twist in the fact the creek actually is used as a side of the rectangle and therefore more fencing is available for the remaining three sides. So let's dive in...

First, we need to record the given information provided:

380 total feet of fencing

3 sides of rectangle to fence.

Ok, so we have two equations which we can bring to bear. The area and the perimeter of the rectangle...

Let's define the length of the rectangle as L and the width of the rectangle as W. So...

The perimeter of the rectangle is 2*L + 2*W (the outside edges), but one of those edges is the creek, so we can remove it from the total perimeter that is associated to the fencing.

Therefore, we have L + 2*W = 380 feet

In addition the Area of the rectangle --> A = L*W

Now we have two equations with two unknowns which we can solve...

Let's solve for the L in the perimeter equation and plug that into the area equation.

380 - 2*W = L

A = (380 - 2*W)*W = 380W - 2W²

We can find the maximized area by either taking the derivative w.r.t. W or looking at the graph and finding the largest point on the graph (note the parabola opens downward!).

Doing it the calculus way --> A'(W) = dA/dW = 380 - 4W = 0 (critical point -- maximum)

dA/dW = 0 = 380 - 4W --> 380 = 4W --> W = 95 feet

With the value of W established, we can then find L --> 380 - 2*W = L --> L = 190 feet

So the area of the largest rectangle with 380 feet of fencing with one side used by the creek is 95 * 190 = 18050 ft²