Final answer:
The maximum area of a garden with 25 feet of fencing is achieved when the pen is a square. Using optimization techniques, the side length is calculated to be 25/4 feet, leading to a maximum area of (25/4)^2 square feet.
Step-by-step explanation:
To find the maximum area of a garden with 25 feet of fencing, we can apply the principles of optimization. Since we need a rectangular pen, let's denote the length as L and the width as W. According to the problem, we have a perimeter P of 25 feet, so the equation for the perimeter is 2L + 2W = 25. To find the maximum area, A = L × W, we need to express one variable in terms of the other using the perimeter constraint.
Let's express W in terms of L: W = (25 - 2L)/2. Now, we can write the area as a function of L: A(L) = L × ((25 - 2L)/2). To maximize this area, we take the derivative of A(L) with respect to L and set it to zero to find the critical points. Upon solving, we find that the maximum area occurs when L is equal to W, meaning the pen is a square. This leads us to an optimal side length of 25/4 feet, resulting in a maximum area of (25/4)^2 square feet.