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Chapter 3: Problem 79

The perimeter of a rectangle is 50 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 114 square feet.

Short Answer

Expert verified
The lengths of the side of the rectangle should be in the range that satisfies the inequality derived by setting the area of the rectangle to less than or equal to 114 square feet, while maintaining the perimeter of the rectangle as 50 feet and ensuring that the length is greater than or equal to the width.

Step by step solution

01

Represent the lengths and widths

Let the length of the rectangle be \(x\), hence the width can be expressed in terms of \(x\) as \( (25 - x) \) since the perimeter is 50 feet.
02

Formulate the area equation

The area of the rectangle should be less than or equal to 114 square feet. This can be represented by the inequality \((x)*(25 - x) \leq 114 \).
03

Solve the inequality

Solve the inequality \( x*(25 - x) \leq 114 \), this will provide the range for the possible lengths of the rectangle.
04

Check the length

From the solution of the inequality, check the possible lengths of the rectangle. Remember, in a rectangle, the length should be more than or equal to the width, which means any value of \( x \) fulfilling the inequality and where \(x \geq 25 - x\) are the possible lengths of the rectangle.

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