Final answer:
The maximum area the farmer can enclose with 150 feet of fencing, using the barn wall for one side, is 2812.5 square feet, calculated by setting the derivative of the area function to zero.
Step-by-step explanation:
The farmer with 150 feet of fencing wants to build a rectangular pen against his barn, utilizing the barn wall as one side of the pen. To find the maximum area that can be enclosed, we need to understand that the maximum area for a given perimeter is achieved with a square shape. However, because one side is the barn wall, we will have a rectangular pen where two sides will be equal (the width), and one side will be the length that will be against the barn.
Let's denote the width of the pen as w and the length as l. The farmer's fencing will cover 3 sides (2 widths and 1 length), so we have 2w + l = 150. The area of the pen will be A = w × l. To maximize the area, we need to express l in terms of w and differentiate the area with respect to w to find the optimal width and length.
Using the fencing equation, l = 150 - 2w. Substituting into the area formula gives us A(w) = w(150 - 2w) = 150w - 2w². To find the maximum area, we take the derivative of A with respect to w and set it equal to zero: A'(w) = 150 - 4w = 0. Solving for w gives us w = 37.5 feet. The maximum length l would then be 150 - 2× 37.5 = 75 feet.
Therefore, the maximum area that the farmer can enclose with 150 feet of fencing, using the barn wall as one side of the pen, is A = 37.5 feet × 75 feet = 2812.5 square feet.