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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 possible lengths of the sides of the rectangle (to not exceed an area of 114 square feet) lie within [9, 14] feet for length and [11, 16] feet for the width.

Step by step solution

01

Understand the perimeter

The perimeter of a rectangle is given by 2(length + width). So, from 2(length + width) = 50, we can simplify this equation to length + width = 25. Suppose, let length = x, so width = 25 - x.
02

Set up the inequality for the area

The area of the rectangle must not exceed 114 square feet, i.e., length*width 鈮� 114. Substituting for width in the area inequality gives the inequation x * (25 - x) 鈮� 114.
03

Solve the inequality

Rearrange the inequality into a quadratic, resulting in \(x^2 - 25x + 114 \leq 0\). This quadratic factors into \((x - 9)(x - 14) \leq 0\). According to the properties of quadratic functions, the function is less than or equal to zero between its roots. So, the possible range of x (length) lies in the interval [9, 14]. The width, therefore, would be in the interval [25 - 14, 25 - 9], which corresponds to the interval [11, 16].
04

Conclude

The possible lengths of the sides of the rectangle, so as not to exceed an area of 114 square feet, should be within [9, 14] feet for length and [11, 16] feet for width.

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