Question

Farmer Bob is enclosing a rectangular pen for his pigs. He will use the side of the barn as one of the longer sides of the pen. He has 88 feet of fencing. What range of lengths of the pen will yield an area greater than 960 square feet?

Answer 1

The range of lengths for the rectangular pen that will yield an area greater than 960 square feet is 16 feet^2_1 < x < 44 feet^2_1.

Step-by-step explanation:

To determine the range of lengths for Farmer Bob's rectangular pen, we start by setting up the problem based on the given constraints. Let x represent the length of the pen perpendicular to the barn.

The total amount of fencing used is the sum of the two sides perpendicular to the barn (2x) and the two sides parallel to the barn (88 feet): 2x + 88 feet.

The formula for the area (A) of a rectangle is length × width. In this case, the length is x, and the width is 2x + 88. So, the area function is A = x(2x + 88).

The problem states that the area should be greater than 960 square feet. Therefore, we set up the inequality:

x(2x + 88) > 960.

Next, we simplify the inequality:

2x^2 + 88x > 960.

By rearranging terms and subtracting 960 from both sides, we get:

2x^2 + 88x - 960 > 0.

Now, we factor the quadratic expression:

(x - 16)(2x + 60) > 0.

Setting each factor greater than zero and solving for x, we find two critical points: x < 16 or x > -60/2.

However, the negative solution doesn't make sense in the context of the problem, so we discard it. Thus, the valid range for x is x < 16.

Therefore, the range of lengths for the rectangular pen is 16 feet^2_1 < x < 44 feet^2_1.