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Chapter 11: Problem 78

The perimeter of a rectangle is 180 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 800 square feet.

Short Answer

Expert verified
The possible lengths of a side of the rectangle are between 20 feet and 40 feet inclusive.

Step by step solution

01

Express Width in terms of Length

Let's denote length as L and width as W. From the perimeter formula, and given that the perimeter is 180 feet, we get \(2*(L + W) = 180\). Bringing it to a form \(W = (180 / 2) - L\), we express the width in terms of length.
02

Substituting W in the Area Formula

Now we will substitute the formula of W into the area formula to get \(A = L*((180 / 2) - L)\). and simplify it to \(A = 90L - L^2\).
03

Setting the Constraints on Length

We were told that the area A must not exceed 800 square feet. So we set up the inequality \(90L - L^2 <= 800\). By rearranging the terms, we get \(L^2 - 90L + 800 <= 0\). Solving this quadratic inequality will give us the possible lengths for L.
04

Solving the Quadratic Inequality

The quadratic inequality can be factored as \((L - 40) * (L - 20) <= 0\). Thus, the solutions for the inequality are 20 and 40. So the possible lengths of a side fall between 20 feet and 40 feet inclusive.

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