Question

Ben has 2000 feet of fencing available to enclose a rectangular field. One side of the field lies along a river, so only three sides require fencing. Express the area A of the rectangle as a function of x, where x is the length of the side perpendicular to the river.

For what value of x is the area largest?

Answer 1

To express the area of the rectangle as a function of x, assume the length perpendicular to the river is 'x' and the widths adjacent to the river are (2000 - x)/2. The area function is A = x * (2000 - x)/2. The value of x that maximizes the area is 500 feet.

Step-by-step explanation:

To express the area of the rectangle, we need to find the length and width of the rectangle in terms of 'x'. Let's assume the length of the rectangle perpendicular to the river is 'x' feet. The two widths of the rectangle adjacent to the river will each be half of the remaining available fencing. So, the widths will be (2000 - x)/2 feet.

The area of the rectangle is the product of its length and width, so A = x * (2000 - x)/2. To find the value of x that maximizes the area, we can take the derivative of A with respect to x, set it equal to zero, and solve for x. Taking the derivative and solving for x, we find x = 500 feet.

Therefore, for an area maximum, the length of the side perpendicular to the river should be 500 feet.