Question

A farmer creates a rectangular pen by using one side of a barn as one side of the pen and using fencing for the other three sides. The farmer has 80 ft of fencing, and the side of the barn is 40 ft long. If x represents

the length of the fenced side of the pen that is parallel to the barn, then the length of each of the two fenced
80-x/2 = 40-0.5x ft. For what values of x is the area sides of the pen that are perpendicular to the barn is
of the pen at least 600 ft²?

Answer 1

600 ft² are x ≤ 111.02.

Explanation:

To find the values of x for which the area of the pen is at least 600 ft², we can start by expressing the area of the pen in terms of x.

The area of the pen is equal to the product of the lengths of the two sides that are perpendicular to the barn. From the given information, we know that the length of each of these sides is 80 - x/2 ft.

Therefore, the area A(x) of the pen is given by:

A(x) = (80 - x/2) * (80 - x/2)

To find the values of x for which the area is at least 600 ft², we can set up the following inequality:

A(x) ≥ 600

(80 - x/2) * (80 - x/2) ≥ 600

Expanding the equation, we have:

(80 - x/2)^2 ≥ 600

Taking the square root of both sides, we get:

80 - x/2 ≥ √600

Simplifying, we have:

80 - x/2 ≥ 24.49

Subtracting 80 from both sides, we obtain:

-x/2 ≥ -55.51

Multiplying both sides by -2 (and flipping the inequality sign), we get:

x ≤ 111.02

Therefore, the values of x that satisfy the condition and give an area of at least 600 ft² are x ≤ 111.02.