Rational Expressions

Question

A farmer with 1200m of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

Answer 1

Each pen would have 1200 ÷ 4 = 300 m of fence, so 2L + 2W = 300 reduced L + W = 150, so L = 150 -W

Area, A, equals L XW, substitute 150 - W for L So A = 150 - W x W given you a quadratic function

-W² + 150W, use x = -b/2a to get x coordinate of vertex (h,k) [maximum]

-150/2(-1) = 75 = max W so L = 150 - 75 = 75

max area of 1 pen = 75 x 75 = 5,625 m²

max area of all 4 pens = 4 x 5625 = 22,500 m²

Answer 2

Well if the long side of the rectangle is r and the other side is s, then the total length of fence is

2r + 2s + 3s = 2r + 5s (if the side length s divides the pens)

2r + 5s = 1200

Area = rs

s = (1200 - 2r)/5 = 0.2 (1200-2r)

Area = 0.2r (1200-2r)

Plot this on Desmos and then find the value of r giving the largest area