Question

An overnight delivery service will not accept any package whose combined length and girth (perimeter of a cross section) exceeds 132 inches. Suppose that you are sending a rectangular package that has square cross sections. If the length of the package is 68 inches, what is the maximum width of the sides of its square cross sections?

Short answer

The maximum width of the sides of the square cross sections is 16 inches.

Step-by-step solution

Understand and Set Up the Inequality

First we understand the problem. We know that the sum of the length and girth (perimeter of the square cross-section) of the package must be less than or equal to 132 inches. So, we can express this condition as an inequality: \(L + 4W \leq 132\). Since the length (L) is already given as 68 inches, we replace L with that value, resulting in the inequality: \(68 + 4W \leq 132\).

Solve the Inequality

Now, we will solve this inequality to find the maximum possible value for the width. To isolate the term '4W', we will subtract 68 from both sides of the inequality: \(4W \leq 132 - 68\). Simplifying the right side, we get: \(4W \leq 64\).To solve for W, we will divide both sides by 4: \(W \leq 64 / 4\).

Obtain the Solution

When we divide 64 by 4, we will get the answer to our inequality: \(W \leq 16\). This means the maximum possible width for the sides of the square cross-sections is 16 inches.

Key concepts

Rectangular Package Calculations

When dealing with rectangular packages, it's essential to understand their geometry. In this context, we focus on packages with square cross-sections. In simpler terms, imagine a package shaped like a long rectangular box where the ends are perfect squares. Calculations often revolve around the dimensions of these cross-sections as they have direct implications for determining conditions like shipping requirements and limitations.
- For a rectangular package with length (\(L\)) and width (\(W\)), the focus is on these measurements to solve problems related to size and fit.
- The square cross-section implies that each of the four sides of the end of the package is of equal length, known as "width" here.
This understanding helps set up equations or inequalities, making it possible to find unknowns like width, given specific perimeter constraints.

Perimeter Constraints

Perimeter constraints are crucial in determining whether a package meets specific size criteria, such as those set by shipping services. These constraints often involve an allowance for the combined lengths and the girth (perimeter of the cross-section) of the package.
- Girth in this context means the perimeter of the square cross-section, calculated as \(4W\) where \(W\) is the side of the square.
- The constraint is formulated as an inequality, often described as a maximum allowable dimension. For the problem at hand, the condition is \(L + 4W \leq 132\). This expression ensures that the combined length and perimeter do not exceed 132 inches.
Understanding these constraints allows for combining dimensional analysis with practical applications ensuring packages are accepted by delivery services.

Maximum Width Determination

Determining the maximum width of a package involves solving the inequality derived from the perimeter constraint. Here’s how to approach it:
- Start by substituting the given length into the inequality: \(68 + 4W \leq 132\). This focuses on isolating \(W\).
- Simplify the inequality step-by-step: subtract the known length (68) from the total allowed combined measurement (132), leading to \(4W \leq 64\).
- Dividing by 4 isolates \(W\), giving us \(W \leq 16\).

This indicates that the maximum side length for each square cross-section is 16 inches. By understanding this process, you can efficiently calculate maximum dimensions for various constraints and ensure all packaging meets required rules.