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Chapter 2: Problem 57

The U.S. Postal Service requires that any box sent by Priority Mail's havelength plus girth (distance around) totaling no more than 108 in. The maximumvolume that meets this condition is contained by a box with a square end 18in. on each side. What is the length of the box? What is the maximum volume?

### Short Answer

Expert verified

Length = 36 in, Volume = 11664 cu in

## Step by step solution

01

## Understanding the Problem

Given a box with a square end 18 in. on each side, we need to find the length of the box that maximizes its volume, under the constraint that the total length plus girth is no more than 108 inches. Girth is calculated as the perimeter of the cross-section perpendicular to the length.

02

## Calculate the Girth

The square end has sides of 18 inches. The girth is the perimeter of this square end. As the girth is the sum of all four sides, calculate it using:\[ \text{Girth} = 4 \times \text{side length} \ \text{Girth} = 4 \times 18 = 72 \text{ inches} \]

03

## Calculate the Maximum Length

Using the constraint length plus girth ≤ 108 inches, solve for length, where girth is 72 inches:\[ \text{Length} + 72 = 108 \ \text{Length} = 108 - 72 = 36 \text{ inches} \]

04

## Calculate the Maximum Volume

The volume of the box is the product of its length and the area of its square end. Calculate the volume using:\[ \text{Volume} = \text{side length}^2 \times \text{length} \ \text{Volume} = 18^2 \times 36 = 324 \times 36 = 11664 \text{ cubic inches} \]

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Maximum Volume Calculation

When dealing with boxes, the goal is often to find the maximum volume that can fit within a given constraint. For this particular problem, the US Postal Service has a rule that the length plus the girth of a box cannot exceed 108 inches. To maximize the volume of the box under this constraint, we need to find the length that makes this possible. This problem is pivotal in packaging and shipping, as businesses want to maximize what they can send while complying with shipping regulations.

To compute the maximum volume, the core idea is to use the constraint to determine the allowable dimensions and then use these dimensions to calculate the volume.

###### Constraint Optimization

Constraint optimization involves finding the best possible solution within a set of restrictions. In our box problem, the USPS constraint is that the length plus the girth of the box should not be more than 108 inches. Here, the girth is the perimeter of the square end of the box.

To find the optimal length, we perform the following steps:

- Calculate the girth by summing up the side lengths of the square end.
- Subtract this girth from the maximum allowed length plus girth to find the remaining length.

These steps ensure you remain within the given constraints while finding the dimension that allows for the largest volume.

Constraint optimization often applies to various real-world scenarios beyond packaging, including resource allocation and design engineering.

###### Girth Calculation

Girth, in this context, refers to the perimeter of the box's cross-section perpendicular to its length. For a box with a square end, the girth is simply four times the length of a side of the square. Here’s the step-by-step way to compute it:

- Identify the side length of the square end (18 inches in this problem).
- Multiply this side length by 4 to get the girth.

So, \[ \text{Girth} = 4 \times 18 = 72 \text{ inches} \]. This calculation is crucial because it determines how much available length is left for the box while still adhering to the total maximum of 108 inches.

Girth is a common measurement in various fields, including packaging and shipping, where it helps in determining the acceptable size of parcels.

###### Box Volume Formula

To find the volume of a box, use the formula for volume, which is the product of the area of the base and the height (or length). For a box with a square base, the formula simplifies to: \[ \text{Volume} = \text{side length}^2 \times \text{length} \]. Using our given dimensions:

- First, calculate the area of the square base (\[ 18 \text{ in.} \times 18 \text{ in.} = 324 \text{ square inches }\]).
- Then, multiply this area by the length of the box (36 inches) to get the volume.

\[ \text{Volume} = 324 \text{ square inches} \times 36 \text{ inches} = 11664 \text{ cubic inches} \]. Understanding this volume formula is essential for numerous applications, including shipping, storage, and product design, where maximizing the use of space is crucial.

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