Problem 57 The U.S. Postal Service requires... [FREE SOLUTION] (2024)

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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, 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.

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.