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Chapter 3: Problem 80

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 lengths of the sides of the rectangle are between 20 feet and 70 feet each.

Step by step solution

01

Write down the equations

Let's denote the sides of the rectangle by length (L) and width (W). The equation corresponding to the given perimeter is \(2*(L + W) = 180\), which simplifies to \(L + W = 90\). The equation corresponding to the area is \(L * W <= 800\).
02

Express one variable in terms of the other

From the perimeter equation, we can express W as \(W = 90 - L\).
03

Substitute the value of W into the area equation

The area constraint becomes \(L * (90 - L) <= 800\). This simplifies into a quadratic inequality \(L^2 - 90L + 800 <= 0\).
04

Solving the quadratic inequality

The roots of the corresponding equation \(L^2 - 90L + 800 = 0\) can be found by using the quadratic formula, yielding \(L = 20 and 70\). With these roots, the solutions of the inequality are \(20 <= L <= 70\).
05

Determining the possible side lengths

Given that \(20 <= L <= 70\), and knowing that \(W = 90 - L\), the possible lengths of the sides of the rectangle are such that \(20 <= W, L <= 70\) with the constraint that \(L+ W = 90\).

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