Calculate the volume and surface area of a standard cereal box. Calculate the cost of manufacturing a standard cereal box if cardboard costs $0.05 per square inch. Design a new cereal box that will hold the same amount of cereal but reduce manufacturing costs. Prove that your new design holds the same amount but can be manufactured more cheaply. Draw a net of your new design. Include all relevant measurements so that your cereal box can be mass produced.
To calculate the volume and surface area of a standard cereal box, we first need to determine the dimensions of the box. A standard cereal box typically has dimensions of 8 inches in length, 4 inches in width, and 12 inches in height.
Volume:
Volume = length x width x height
Volume = 8 in x 4 in x 12 in
Volume = 384 cubic inches
Surface Area:
Surface Area = 2lw + 2lh + 2wh
Surface Area = 2(8in x 4in) + 2(8in x 12in) + 2(4in x 12in)
Surface Area = 64 + 192 + 96
Surface Area = 352 square inches
Manufacturing cost:
Cost = surface area x cost per square inch
Cost = 352 sq in x $0.05
Cost = $17.60
For the new cereal box design, we can reduce costs by reducing the amount of material used. One way to do this is to make the box rectangular instead of square, while still maintaining the same volume.
Let the dimensions of the new box be:
Length = 8 inches
Width = 4 inches
Height = 6 inches
Volume:
Volume = 8 in x 4 in x 6 in
Volume = 192 cubic inches
Surface Area:
Surface Area = 2lw + 2lh + 2wh
Surface Area = 2(8in x 4in) + 2(8in x 6in) + 2(4in x 6in)
Surface Area = 64 + 96 + 48
Surface Area = 208 square inches
Manufacturing cost:
Cost = surface area x cost per square inch
Cost = 208 sq in x $0.05
Cost = $10.40
Therefore, the new design holds the same amount of cereal but can be manufactured more cheaply. Here is the net for the new cereal box design:
```
___________
/ /|
8in / / |
/ / |
/__________/ 4in
| | /
| | / 6in
| |/
|__________|
```