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Chapter 2: 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 a side of the rectangle can range from approximately 5.14 feet to 19.86 feet, so that the area of the rectangle will not exceed 114 square feet.

Step by step solution

01

Solve for Width

To solve for the width (W), use the formula for the perimeter of a rectangle, which is \(2(L + W) = 50\) feet. This implies that \(W = (50 - 2L)/2\).
02

Substitute for Width in Area Formula

Substitute \(W\) in the formula for the area of rectangle, which is \(A = L*W\), to obtain \(A = L*((50 - 2L)/2)\). Simplify this to get \(A = 25L - L^2\)
03

Validate the Area Limitation

Given that the Area should not exceed 114 square feet, this means that \(25L - L^2 <= 114\). If simplified, we obtain a quadratic equation \(L^2 - 25L + 114 >= 0\). Solve this equation for \(L\) to obtain the possible lengths of a side.

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