Question

UPS defines the size of a box to be equal to its length plus twice its width plus twice its height: size = length + 2 * width + 2 * height. In order to avoid being charged freight charges, the size of a package has to be less than 165 and the length has to be less than 108. You work for a wine producer who ships wine bottles in rectangular boxes that are 13 x 4 x 4. The individual boxes are packaged into larger rectangular boxes for shipping. In order to go through the company's production line, the length of the boxes has to be 13. The company wants to minimize its shipping costs so they need to send as few of the large boxes as possible. What size box will hold the maximum number of the smaller boxes?

(a) 13 x 42 x 42
(b) 13 x 49 x 49
(c) 13 x 52 x 52
(d) 13 x 54 x 54

Answer 1

The question pertains to finding the appropriately sized shipping box for wine bottles to minimize shipping costs within UPS size restrictions using the formula size = length + 2*width + 2*height, with specifics given for the smaller wine boxes.

Step-by-step explanation:

The student is asking a Mathematics problem related to optimizing the size of packaging to minimize shipping costs based on UPS's size specifications. UPS's formula for calculating the size of a package is size = length + 2*width + 2*height. The constraints are that the size must be less than 165 and the length must be less than 108. The smaller wine boxes are 13 inches in length and 4 inches in both width and height. We need to find the size of the larger box that can hold the maximum number of smaller boxes without exceeding the size limit set by UPS.

To solve this, we apply the UPS formula to each box option and check which one maximizes the number of smaller boxes while staying within the size restriction. For example, option (a) calculates as size_a = 13 + 2*42 + 2*42 = 181, which exceeds the UPS limit. The calculation process should be applied to each shipping option (b, c, and d) to identify the one that meets the criteria while allowing for the maximum packing of smaller wine boxes.