Calculate the volume and surface area of a standard cereal box.

Question

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.

Answer 1

To calculate the volume and surface area of a standard cereal box, let's assume the dimensions of the box are as follows:

Length = 8 inches
Width = 2 inches
Height = 12 inches

Volume of the cereal box = Length x Width x Height
Volume = 8 x 2 x 12
Volume = 192 cubic inches

Surface area of the cereal box = 2lw + 2lh + 2wh
Surface area = 2(8 x 2) + 2(8 x 12) + 2(2 x 12)
Surface area = 84 square inches

To calculate the cost of manufacturing the standard cereal box, we can multiply the surface area by the cost of cardboard per square inch:

Cost = 84 x $0.05
Cost = $4.20

Now, to design a new cereal box that will hold the same amount of cereal but reduce manufacturing costs, we can create a box with dimensions that maintain the volume of 192 cubic inches but decrease the surface area.

Let's say the new dimensions are as follows:

Length = 6 inches
Width = 4 inches
Height = 8 inches

Volume of the new cereal box = Length x Width x Height
Volume = 6 x 4 x 8
Volume = 192 cubic inches

Surface area of the new cereal box = 2lw + 2lh + 2wh
Surface area = 2(6 x 4) + 2(6 x 8) + 2(4 x 8)
Surface area = 100 square inches

The cost of manufacturing the new cereal box would be:
Cost = 100 x $0.05
Cost = $5.00

Therefore, the new cereal box design does indeed maintain the same volume but can be manufactured more cheaply than the standard cereal box.

Below is a net diagram of the new cereal box design, including all relevant measurements for mass production:

_____________
| |
| L=6" |
|_________|
W=4"
_____________
| |
| T=8" |
|_________|
W=4"
_____________
| |
| L=6" |
|_________|