College Algebra application/ function problem?

Question

How do I solve this problem?

Beth has 3000 ft 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 the largest?
c)what is teh maximum area?

Answer 1

Since Beth has 3000 ft of fencing, and the enclosing requires a length and a width (which we'll denote x and y), then we know that

2x + 2y = 3000 (there are two lengths and two widths for the field, since it's rectangular).

We can reduce that to x + y = 1500.

The area is then expressed as

A = xy

But x + y = 1500, or y = 1500 - x, so

A = x(1500-x) = 1500x - x^2

So our function is then
A(x) = 1500x - x^2

(b)
To solve for what value of x is the area the largest, we have to take the derivative, make it 0, and solve for x.
A'(x) = 1500 - 2x
0 = 1500 - 2x
2x = 1500
x = 750

(c) To get the maximum area, we just plug in x=750 for the function we just created.
A(750) = 1500(750) - (750)^2

Answer 2

a) A=x(1500-x)
b) x=750 feet
c) Maximum area = 562,500 square feet

If it helps, in these "maximum area" problems where the fencing encloses a rectangle, the maximum area is when the length and width are equal; in other words when it is a square.