Question

A dog trainer is installing a small rectangular pen using at most 40 feet of fencing for three sides. The fourth side of the pen will be against the wall of a building. She needs at least 182
square feet of space. What are the possible lengths x of the shorter side of the pen? Write the answer in inequality notation.

Answer 1

The 40 feet of fencing which covers at least 182 square feet gives the possible lengths of x as x ≥ 13 or x ≤ 7

How can the area be expressed as an inequality?

Length of the fencing = 40 feet

Number of sides covered = Three sides

Area of the fencing ≥ 182 square feet

Length of the shorter side of the pen = x

The area is therefore;

• x × (40 - 2•x) ≥ 182

2•x² - 40•x + 182 ≥ 0

2•(x² - 20•x + 91) ≥ 0

2•(x² - 13•x - 7•x + 91) ≥ 0

2•(x•(x - 13) - 7•(x - 13) ≥ 0

• 2•(x - 13)•(x - 7) ≥ 0

Which gives;

• x ≥ 13 or x ≤ 7

The solution to the inequality and the possible lengths of x are; x ≥ 13 or x ≤ 7

Learn more about quadratic inequalities here;

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