Question

A company that manufactures gift boxes wants to know how many different-sized boxes having a volume of 50 cubic centimeters it can make if the dimensions must be whole centimeters.

a. List all the possible whole number dimensions for the box.
b. Which possibility requires the least amount of material to make?
c. Which box would you recommend the company use? Why?

Answer 1

a) Box1: 1 cm × 1 cm × 50 cm

Box2: 1 cm × 25 cm × 2 cm

Box3: 1 cm × 5 cm × 10 cm

Box4: 5 cm × 5 cm × 2 cm

b) Box 4 needs lees material. S=90cm²

c) I would recommend to use the 5 cm×5 cm×2cm box because it uses less material.

Explanation:

a) The number 50 can be represented as:

50=1×1×50

50=1×5×10

50=1×25×2

50=5×5×2

Hence, these are all the possibilities of the dimensions of the boxes:

Box1: 1 cm × 1 cm × 50 cm

Box2: 1 cm × 25 cm × 2 cm

Box3: 1 cm × 5 cm × 10 cm

Box4: 5 cm × 5 cm × 2cm

b)
`S_(Box1)=2(1cm*1cm)+2(1cm*50cm)+2(50cm*1cm)=201 cm^2`

`S_(Box2)=2(1cm*25cm)+2(25cm*2cm)+2(25cm*1cm)=200cm^2`

`S_(Box3)=2(1cm*5cm)+2(5cm*10cm)+2(10cm*1cm)=130cm^2`

`S_(Box4)=2(5cm*5cm)+2(5cm*2cm)+2(5cm*2cm)=90cm^2`