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 find the largest volume of the package, we need to determine the dimensions of the package that maximize the volume while still satisfying the given constraints of the sum of the longest side and the girth.

Let's assume the square ends of the package have sides of length "x".

Given that the length of the package is 40in, the sum of the length of the longest side and the girth can be calculated as:

Longest side = length = 40in
Girth = 2 * (x + x) = 4x

Therefore, the sum is: 40in + 4x

According to the given constraints, the sum should be greater than 84in and less than or equal to 108in.

84in < 40in + 4x ≤ 108in

Now, let's solve the inequality to find the range of possible values for "x":

84in < 40in + 4x ≤ 108in
Subtracting 40in from all sides:
44in < 4x ≤ 68in
Dividing all sides by 4:
11in < x ≤ 17in

So, the possible range for the length of one side of the square ends is 11in < x ≤ 17in.

To find the largest volume, we need to maximize the length of "x". Since we want square ends, the length of each side should be the same, which means "x" will be equal to one side of the square.

Therefore, to maximize the volume, we choose the maximum value of "x" from the given range, which is x = 17in.

To calculate the volume of a rectangular prism (the package), we multiply the length, width, and height. Since the package has square ends, the volume can be expressed as:

Volume = length × width × height
Volume = 40in × 17in × 17in

Calculating the volume:

Volume = 40in × 17in × 17in
Volume = 115,600 cubic inches

Therefore, the largest volume the package can have is 115,600 cubic inches, and the dimensions of the package with the maximum volume are:
Length = 40in
Width = 17in
Height = 17in