We are tasked with the problem of constructing a cardboard b

We are tasked with the problem of constructing a cardboard box that will perfectly fit 270 1cm times 1cm times 1cm cubes. 1. Find all possible dimensions for the box. Be sure to explain how you know you\'ve found them all. 2. Once the cubes are in the box, we plan on wrapping the box with gift wrap before shipping it to our customers. Unfortunately, gift wrap is relatively expensive, so we want to use as little as possible. Determine how much gift-wrap will be required for each of the box dimensions you found in (1) and determine the most cost-effective dimensions of the box.

Solution

number of boxes needed =270

each box diamention =1cm^3

so volume of cardboard=270cm^3

270=3*5*3*2*3 we need l,b,h combination (length ,breadth, height)

l,b,h

270=15*3*6

270=15*2*9

270=15*9*2

270=10*3*9

270=30*3*3

270=45*3*2

270=5*9*6

270=5*3*18

these are the possible combinations

2) here we have sides with area

lb,bh,lh,lb,bh,lh

find area of each side and take sheets as per the area

first case l=15 , b=3,h=6.

lb=45cm^2 bh=3*6=18cm^2 lh=15*6=90cm^2

2 sides with 45cm^2 and 2 sides with 18cm^2 and 2 sides with 90 cm^2