Area, dimensions

Question

600 m of fencing is used to construct 6 rectangular animal pens as shown.

a Show that the area A of each pen is A= (-8/9)x^2 + (200/3)x * m^2

b Find the dimensions of each pen so that it has the maximum possible area.

c What is the area of each pen in this case?

Answer 1

600 feet fencing, 6 rectangular pens

without seeing a diagram, it may help to draw a rought sketch of 6 pens adjoining each other

that makes 7 fences in one direction and 2 in the other

7w +2L = 600

2L = 600-7w

L = 300-7w/2

Area = A = Lw = (300-7w/2)w = 300w -7w^2/2

A' = 300-7w = 0

w = 300/7 = 22 6/7 meters wide

L = 300-7(300/7)/2 = 300-150= 150 meters Long

7(300/7) + 2(150) = 300+300 = 600 feet fencing

max Area = 150(300/7) =45000/7 = 6,428 4/7 ft^2

each of the 6 pens then is 22 6/7 by 25 meters

with area = 45000/42 = 1,071 3/7 square meters