A farmer has 1,000 yards of fence to use to make a rectangular corral. Let x be the length of the corral.
Why does A(x)=x(500-x) represent this situation?
Where did the 500 come from?
half the perimeter is 500
one side is x, so the other is 500-x
In this situation, the farmer wants to use the fence to create a rectangular corral. The perimeter of a rectangle is the sum of all its sides, which in this case would be 2 sides of length x, and 2 sides of length 500-x (since the total fence length is 1000 yards and two sides will be of length x, the other two sides must be of length 500-x).
The area of a rectangle is found by multiplying its length by its width. In this case, the length is x and the width is 500-x. Therefore, the formula to calculate the area of the corral is A(x)=x(500-x).
The number 500 represents the difference between the total fence length (1000 yards) and one side of the rectangle (x). The equation A(x)=x(500-x) allows us to find the maximum area of the corral by maximizing the product (x * (500-x)) within the constraint (x + 500-x = 1000).