maximun area that can be enclosed

Question

A rectangular field will be fenced on all four sides. Fencing for the north and south sides costs $5 per foot and fencing for the other two sides costs $15 per foot. What is the maximum area that can be enclosed for $5700

Answer 1

2 sides cost 3 times the other 2 sides

that means you want 3 times the length for the cheaper sides, as the more expensive sides

2[5(3x)+15(x)] = 5700

60x = 5700

x = 5700/60 =570/6 = 95 feet for the 2 shorter sides

3x = 285 feet for each of the 2 longer sides

Maximum area enclosed is 95 times 285 = 27,075 square feet

Another way to solve this problem is set up an equation for Area

A = LW

2(5L + 15W) = 5700

10L + 30W = 5700

10L =5700 - 30W

L = 570 -3W

substitute into the area equation

A = (570-3W)W = 570W -3W^2

now take the derivative and set = 0

A' = 570 -6W = 0

6W = 570

W = 95 feet

L= 570-3(95) = 570 - 285 = 285 feet

Max area = 95 x 285 = 27,075 square feet