Question

A farmer uses 1000 meters of fencing to build a rectangular corral. One side of the corral has length x. Express the area A of the corral as a function of x.

Answer 1

A(x) = 500x - x²

Explanation:

The farmer needs 1000m of fencing to build the rectangular corral. You can also see this as the rectangular corral has a perimeter of 1000m. The expression used to find the perimeter of a rectangle is:

p = 2x + 2y

Where p is the perimeter, x is the length of one of the horizontal sides and y is the length of one the vertical sides. Knowing this, we can express y in terms of the perimeter and x, by isolating y:

y = (p - 2x)/2

Now, the area of that rectangle would be:

A = x*y

So we replace the expression we found for y in the expression for the Area and we get:

A = x*y = x*(p-2x)/2 = x*p/2 - x²

Replacing p with its value of 1000, we the Area in terms of x:

A(x) = 1000*x/2 - x² = 500*x - x²