Question

(3 pts) A rancher has 1200 feet of fence material to enclose a rectangular pasture and partition it as

shown.
a) Write an equation in standard form and in vertex form
for the area of the entire pasture as a function of just
one variable.
b) What is the maximum area enclosed?
c) Determine the dimensions of the pasture which would maximize the area enclosed by fence.

Answer 1

Explanation:

Let the width of the field be x fee, then the length will be (1200 - 2x) / 2

= (600 - x) feet

The area:

A = x(600 - x)

A = -x^2 + 600x (Standard Form)

A = - (x^2 - 600))

= -(x - 300)^2 + 9000 (Vertex form)

b) The maximum area is when (x - 300)^2 = 0

That is when x = 300.

So it's 90,000 ft^2.

c) The dimensions for maximum area is

300 ft width and 600 - 300 = 300 ft length.