Question

a company that manufactures and ships canned vegetables is designing boxes in the shape of rectangular prisms to meet specific requirements. The vegetables are packed into cans that are 3 inches in diameter and 5 inches in height. Company regulations state that boxes must be filled in with two layers of 12 cans each, and be completely closed with no overlapping material. What is the smallest amount of cardboard needed to meet the company's requirements

Answer 1

In order to find the smallest amount of cardboard needed, you need to find the total surface area of the rectangular prism.

Therefore, you need to understand how the cans are positioned in order to find the dimensions of the boxes: two layers of cans mean that the height is
h = 2 · 5 = 10 in

The other two dimensions depend on how many rows of how many cans you decide to place, the possibilities are 1×12, 2×6, 3×4, 4×3, 6×2, 12×1. 
The smallest box possible will be the one in which the cans are placed 3×4 (or 4×3), therefore the dimensions will be:
a = 3 · 3 = 9in
b = 3 · 4 = 12in

Now, you can calculate the total surface area:
A = 2·(a·b + a·h + b·h)
   = 2·(9·12 + 9·10 + 12·10)
   = 2·(108 + 90 + 120)
   = 2·318
   = 636in²

Hence, the smallest amount of carboard needed for the boxes is 636 square inches.