Thread: Enclosing a Rectangualr Field

Beth has 3000 feet of fencing available to enclose a rectangular field.

a) Express the area A of the rectangle as a function of x, where x is the length of the rectangle.

b) For what value of x is the area largest?

c) What is the maximum area?

I have the answer but do not know how to set up. Can someone help.

Answers
a) A(x)=-x^2 + 1500x

b) A is largest when X = 750ft.

c) 565,500 ft sq

Robert

One way of doing this is by Analytic Geometry, by the properties of the parabola.
Another way, an easier way, is by Calculus. I will use this way here.

Rectangle.
Length = x ft.
Perimeter = 3000 ft.
So,
Width = (3000 -2x)/2 = 1500 -x ft.

a) Area, A = x(1500 -x) = 1500x -x^2

b) A is largest or smallest when dA/dx = 0
dA/dx = 1500 -2x
Set that to zero,
0 = 1500 -2x
2x = 1500
x = 1500/2 = 750 ft.
That should be the x for maximum A. (because minimum A is zero, and here x=0 or x=1500 ft.)

c) A = 1500x -x^2
max A = 1500(750) -(750)^2 = 562,500 sq.ft.
(Not 565,500 sq.ft.)