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Chapter 2: Problem 73

A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?

Short Answer

Expert verified
The length of the rectangle must be between \(25 - \sqrt{225}\) and \(25 + \sqrt{225}\) meters.

Step by step solution

01

Formulate the perimeter

The formula for the perimeter of a rectangle is \( P = 2 \times (length + width) \). Given that the perimeter is 100, we can say: \( 2 \times (length + width) = 100 \) or \( length + width = 50 \).
02

Solve for the width

From step 1, we can rearrange the formula to solve for the width. \( width = 50 - length \)
03

Formulate the area

The area of a rectangle is length 脳 width. With the width formula from Step 2, the area becomes: \( Area = length \times (50 - length) \)
04

Solve for the length

Given that the area is at least 500, we can set up an inequality: \( 500 \leq length \times (50 - length) \). Solving this quadratic inequality gives the bounds for the length of the rectangle.
05

Solve the inequality

By solving the inequality we get two roots, these roots show us the bounds. It is found that \(25 - \sqrt{225} \leq length \leq 25 + \sqrt{225}\). So the length of the rectangle must be within these bounds

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