calculus exam help

Question

A school is planning to fence off a rectangular playground with three additional fences across its width to have separate playgrounds for different age groups as shown in the diagram below. Find the maximum area of the large rectangular playground that the school can enclose with 120 m of fencing.

Answer 1

Area A = (120 - 8L)L There are 8 L lengths when you divide the big rectangle by 4 equal rectangular segments. See figure at bottom.

A = -8L² + 120L -b/2a is the maximum (vertex) for this parabola = -120/-16 = 7.5

A = -8(7.5²) + 120(7.5) = 450 ft²

There are 2 lengths (L) and 5 widths (W) when you divide the large rectangle into 4 equal sections.

Area = 5W x 2L but the Perimeter = 120 = (5W + 2L), then W = 24 - 0.4L

Then Area = L(24 - 0.4L) = 24L - 0.4L²

Taking the derivative = 0 yields A' = 24 - .8L = 0, and L = 30

Then W = (120 - 60)/5 = 12

The Area = LW = 30x12 = 360 ft²

__L_____L_____L_____L___

| | | | | W

| _L__|__L__ |___L__|__L___|