Question

A company is creating packaging to store cylindrical soda cans. The box they are storing them in has a volume of 1000cm cubed.

If the cans have a radius of 1.7cm and a height of 4.5 cm, how many cans could they store?

PLEASE SHOW YOUR WORK

Answer 1

For cans

• r=1.7cm
• h=4.5cm

Volume:-

• V=πr²h
• V=π(1.7)²(4.5)
• V=2.89(4.5)π
• V=40.8cm³

Total cans

• 1000/40.8
• 24.5
• 24 cans

As you can't put a half can in spare space round to the previous no

Answer 2

Packing in a rectangular prism box

As the cans are cylindrical in shape, they are not able to be packed into a conventional rectangular prism box without wasted space. Therefore, we need to model the cans as rectangular prisms with the following dimensions:

• height = 4.5 cm
• width = diameter = 2 × radius = 2 × 1.7 cm = 3.4 cm
• length = width = 3.4 cm

Volume of a rectangular prism = width × length × height

= 3.4 × 3.4 × 4.5

= 52.02 cm³

Therefore, the volume of space a can will take up in the box is 52.02 cm³ (regardless of the orientation in which they are packed).

To calculate how many cans can be stored in a box with a volume of 1000 cm³, divide the volume of the box by the found volume of the space needed for one can.

Number of cans = volume of box ÷ volume of space needed for one can

= 1000 cm³ ÷ 52.02 cm³

= 19.22337562...

= 19

Therefore, the box can theoretically store a maximum of 19 cans.

Packing in a cylindrical box (tube)

If the box was a tube (cylinder) where each can could be stacked on top of each other, we can model the can as a cylinder:

`\textsf{Volume of a cylinder}=\sf \pi r^2 h \quad\textsf{(where r is the radius and h is the height)}`

⇒ Volume of a can = π (1.7)² (4.5)

= 40.85641246... cm³

Number of cans = volume of box ÷ volume of space needed for one can

= 1000 cm³ ÷ 40.85641246... cm³

= 24.47596203...

= 24

Therefore, a cylindrical tube box can store a maximum of 24 cans.