Enclosing the Largest Area

Question

 The owner of the Rancho Los Feliz has 3480 yd of fencing to enclose a rectangular piece of grazing land along the straight portion of a river and then subdivide it by means of a fence running parallel to the sides. No fencing is required along the river. (See the figure below.) What are the dimensions of the largest area that can be enclosed?

Answer 1

Let's first define the perimeter of the fence: P = 2*width + length = 3480 yd (we don't need 2 lengths because there is no fence along the river).

We can solve for length, such that: length = 3480 - 2*width

Now, let's define the area of the enclosed space: A = width*length = width*(3480 - 2*width) = 3480*width - 2*width²

In order to maximize the enclosed area, we can take the derivative of A, set it equal to 0 and solve for the width: A' = 3480 + 4*width = 0 ----> width = 3480/4 = 870 yd

We can now solve for the length: length = 3480 - 2*width = 3480 - 1740 = 1740 yd

Let's check to make sure our perimeter is correct: P = 2*width + length = 2*870 + 1740 = 3480 (we're good!)

Finally, we can calculate the enclosed area: A = width*length = (1740)(870) = 1,513,800 yd²