Part A
The rectangular prism is a box, and the volume of the box is
V = L*W*H
so we simply multiply the length (L), width (W) and height (H)
In this case, the dimensions of the box are
The order doesn't matter in this case because we can multiply two or more numbers in any order we want (eg: 6*7 = 7*6 = 42)
So,
V = L*W*H
V = 13*5*6
V = 390
The volume of one box is 390 cubic inches.
I'm not sure what formula Diego is using. It seems like he's computing the perimeter of the base (the 13 inch by 5 inch rectangle) to get 13+5+13+5 = 36 inches. Then he's multiplying that by 6 to get 36*6 = 216; however, this value is meaningless because we won't involve the perimeter at all when it comes to finding the volume.
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So in short, Diego is incorrect. All Diego needs to do is multiply the three values 13, 5 and 6 to get the volume of one box to be 390 cubic inches.
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Part B
Use the same formula back in part A. That formula being V = L*W*H.
We'll use this formula to compute the volume of each carton.
- Carton A has a volume of 10*13*28 = 3640 cubic inches
- Carton B has a volume of 10*14*22 = 3080 cubic inches
- Carton C has a volume of 11*12*26 = 3432 cubic inches
All three cartons have a volume larger than 390 cubic inches, meaning that we can fit at least one box.
If one box is 390 cubic inches, then 8 of them lead to a volume of 8*390 = 3120 cubic inches. Note that this value (3120) is larger than the volume of carton B. There's no way that Diego can use carton B to hold all 8 boxes at once. So we have no choice but to rule it out.
Carton C on the other hand has a volume of 3432 cubic inches, which is larger than the 3120 figure we got. Meaning he can house all 8 boxes here. The amount of leftover space is 3432-3120 = 312 cubic inches which is what the packing material will take care of. In other words, you need 313 cubic inches of packing material for carton C. Carton A is a similar story, but will require more packing material (since carton A's volume is larger).
The best option is to go with carton C. It allows him to hold all the 8 boxes, with the least amount of packing material required.