A farmer has 240 feet of fencing and wants to build two identical pens for his prize-winning pigs. The pens will be arranged as shown. Determine the dimensions of a pen that will maximize its area.

Question

Quizplus · Accepted Answer

Let x be the length of one side of the square pen. Then the total length of fencing used is 3x (two sides of length x and one side of length 2x for the shared pen boundary). Thus, 3x = 240, which means x = 80. So each side pen is 80 feet long.

The area of one pen is A = x^2 = 80^2 = 6400 square feet. The total area for both pens is 12800 square feet.

To find the maximum area, we need to use calculus. We take the derivative of A with respect to x:
dA/dx = 2x
We set this equal to zero and solve for x:
2x = 0
x = 0
This is not a valid solution (we cannot have a pen with zero length), so there is no minimum or maximum value. However, since the function dA/dx = 2x is increasing for x > 0 (positive values of x), we know that the area is increasing as x increases up to the maximum possible value of x = 80 (the length of the fencing). Therefore, we conclude that the dimensions of the pen that maximize its area are 80 feet by 80 feet.

A Farmer Has 240 Feet of Fencing and Wants to Build