Problem 40 - Algebra 2

Question

MAIL For Exercises 40 and 41 , use the following information The U.S. Postal Service defines an oversized package as one for which the length \(L\) of its longest side plus the distance \(D\) around its thickest part is more than 108 inches and less than or equal to 130 inches. Write a compound inequality to describe this situation.

Step-by-Step Solution

Answer

The compound inequality is \( 108 < L + D \leq 130 \).

Step 1: Understand the Problem

The problem defines an oversized package based on the measurements of the package's dimensions. Specifically, it involves two measurements: the length of the longest side, denoted as \( L \), and the distance around the thickest part, denoted as \( D \). The combined measurement \( L + D \) needs to exceed 108 inches but be less than or equal to 130 inches to be considered oversized.

Step 2: Construct the Inequality

We need to set up a compound inequality based on the criteria provided. The combined length of \( L + D \) is more than 108 inches, which means \( L + D > 108 \). Additionally, this combination is less than or equal to 130 inches, meaning \( L + D \leq 130 \). Therefore, the compound inequality representing the situation is: \[ 108 < L + D \leq 130. \]

Step 3: Verify the Inequality

Ensuring that this inequality accurately represents the problem statement, we check each condition separately. For oversize, \( L + D \) needs to be greater than 108 and no more than 130. Our inequality \( 108 < L + D \leq 130 \) correctly reflects these requirements.

Key Concepts

Oversized Packages

When dealing with postal services, understanding what qualifies as an "oversized package" is key, especially for anyone looking to ship large items. The U.S. Postal Service terms a package as "oversized" based on two specific dimensions. The criteria involve:

• The length of the package's longest side, referred to as \( L \).
• The distance around the thickest part of the package, referred to as \( D \).

Together, these dimensions must satisfy a particular condition. The sum of \( L \) and \( D \) must be more than 108 inches and less than or equal to 130 inches. Therefore, an oversized package is not just about size but about how the size is distributed across different measurements. This ensures that packages fitting this description are handled appropriately when being shipped and priced.

Geometry

Geometry plays an integral role in understanding the dimensions and shapes of packages. When we discuss the longest side and the thickest part, we are diving into geometric concepts.

• The **longest side**, \( L \), is the maximum length of any edge on the package.
• The **thickest part**, \( D \), involves calculating the 'girth', essentially the circumference around the widest section of the package. Imagine wrapping a tape measure around the thickest part of the package.

This is crucial for deciding how an object fits within certain spatial limitations and ensures it adheres to shipping standards. Understanding these measurements enables accurate classification of packages and efficient space management.

Inequality Solving

The process of inequality solving involves manipulating mathematical statements to find permissible values for variables. In this context, the problem is defined as a compound inequality: \[108 < L + D \leq 130.\]This means that the sum of \( L \) and \( D \) must be greater than 108 inches but not exceed 130 inches. To solve any compound inequality, follow these steps:

• **Understand the boundaries**: Identify the lower and upper limits which are 108 and 130 in this case.
• **Isolate variables if necessary**: For more complex equations, simplify to isolate \( L + D \).
• **Check the solutions**: Ensure that solutions make logical sense within the context; all values for \( L + D \) should fit the original geometric stipulations.

Such inequalities help in setting parameters for measurements ensuring all results follow within the defined constraints.