The post office ships a package using large package rates if the sum of the length of the longest side and the girth (distance around the package perpendicular to its length) is greater than 84in and less than or equal to 108in. Suppose you need to ship a package that is 40in in length and has square ends. What is the largest volume that the package can have? What are the dimensions of that package?

Question

The post office ships a package using large package rates if the sum of the length of the longest side and the girth (distance around the package perpendicular to its length) is greater than 84in and less than or equal to 108in. Suppose you need to ship a package that is 40in in length and has square ends. What is the largest volume that the package can have? What are the dimensions of that package?

Answer 1

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Answer 2

To determine the largest volume and dimensions of the package that can be shipped within the given conditions, we need to find the dimensions (width and height) that maximize the volume while satisfying the restrictions.

Let's call the width and height of the package "w" and "h", respectively.

Given:
Length of the package (longest side) = 40 inches
Perpendicular distance around the package (girth) = 2w + 2h

According to the conditions:
Sum of longest side length and girth > 84 inches
Longest side length + 2w + 2h > 84

Substituting the given values:
40 + 2w + 2h > 84
2w + 2h > 44
w + h > 22

The volume of a rectangular prism is given by:
Volume = length * width * height

Given the length = 40, we need to maximize the volume by finding the largest possible values for width and height.

Since we have a square end, w = h.

Substituting, we have:
2w > 22
w > 11

Now, we need to find an appropriate value for "w" greater than 11 and calculate the corresponding "h".

Let's start with the closest whole number greater than 11 for "w" which is 12.

Substituting w = 12 into the equation w + h > 22:
12 + h > 22
h > 10

This means any value of "h" greater than 10 will satisfy the condition.

Now we can calculate the volume:
Volume = length * width * height
Volume = 40 * 12 * 12
Volume = 5,760 cubic inches

Therefore, the largest volume the package can have is 5,760 cubic inches, and the dimensions of the package with the largest volume are 40 inches long, 12 inches wide, and 12 inches high.

Note: The package dimensions are not unique, and other combinations may also result in the same largest volume.