Final answer:
The maximum area for the pig enclosure is found by optimizing the area function A(x) = 200x - 2x^2 given the perimeter constraint 2x + y = 200, which results in an enclosure of 50 feet by 100 feet with a maximum area of 5000 square feet.
Step-by-step explanation:
The student is asking how to maximize the area of a pig enclosure using 200 feet of fencing with the barn acting as one side of the fence. To find the maximum area, we treat the problem as an optimization problem in calculus, assuming a rectangular shape for the enclosure for simplicity. Let the lengths of the two sides perpendicular to the barn be represented by x, and the length parallel to the barn be y. Since the barn is used as one side, we only need fencing for the other three sides. This situation gives us the equation x + x + y = 200, which simplifies to 2x + y = 200. To express the area A as a function of x, use the area formula A = x × y. To find y, we rearrange the perimeter equation to y = 200 - 2x and substitute it into the area equation, resulting in A(x) = x(200 - 2x). This simplification yields A(x) = 200x - 2x^2. To find the maximum area, take the derivative of A with respect to x and find the critical points. Setting the derivative equal to zero gives us the value of x that maximizes the area.
Upon solving, we find that the maximum area is achieved when x is 50 feet, so the optimum dimensions are 50 feet by 100 feet, leading to the maximum area of 5000 square feet, given by A(50) = 50 × (200 - (2 × 50)) = 5000. Therefore, the correct answer to the student's question is: a) Equations needed to maximize area.