A farmer is building a rectangular pen with three parallel partitions, shown in the diagram. He has 500 feet offencing available. What is the maximum area he can enclose?
Accepted Solution
A:
Answer:10,416.67 ft²Step-by-step explanation:If the length of the three parallel partitions is h, and the width of the two parallel walls is w, then the perimeter is:500 = 3h + 2wThe area is:A = hwSolve for w in the first equation:2w = 500 − 3hw = 250 − 1.5hSubstitute into the second equation:A = h (250 − 1.5h)A = 250h − 1.5h²We can find the maximum using vertex of a parabola:h = -b / (2a)h = -250 / (2 × -1.5)h = 83.33Or using calculus:dA/dh = 250 − 3h0 = 250 − 3hh = 83.33Therefore, the maximum area is:A = 250h − 1.5h²A = 10,416.67