Final answer:
To find the dimensions of the pen with the maximum area, we need to maximize the area of the rectangular pen by solving for W and L. The dimensions of the pen with the maximum area are approximately 31.25ft by 31.25ft (option C).
Step-by-step explanation:
To find the dimensions of the pen with the maximum area, we need to maximize the area of the rectangular pen. The total amount of fencing available is 125 feet. Let's assume the dimensions of the rectangular pen are length (L) and width (W).
The perimeter of the pen can be expressed as: 2L + W = 125 (since one side is bordered with the barn).
Now, we can express the area of the pen as: A = L * W.
To solve for the maximum area, we can use a mathematical technique called optimization. Solving the perimeter equation for L, we get: L = (125 - W) / 2.
Substituting this value of L into the area equation, we have: A = ((125 - W) / 2) * W.
We can now find the dimensions of the pen with the maximum area by determining the value of W that maximizes the area. To do this, we can take the derivative of A with respect to W, set it equal to zero, and solve for W. After finding W, we can substitute it back into the perimeter equation to find L.
The dimensions of the pen with the maximum area are approximately 31.25ft by 31.25ft (option C).