Question

Consider the relationship between surface are and volume of the three-dimensional figures you have learned about and

select the one that you would like to use for your cereal box design. Write a paragraph discussing the figure you will use
for your packaging design, including:
 the name of the figure you chose
 the formulas for its surface area and its volume
 the reasons why you believe it is the best figure for your cereal box

Answer 1

[AI-generated answer — Bot GPT 3.5]

I have chosen to use a rectangular prism for my cereal box design. The surface area formula for a rectangular prism is SA = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height. The volume formula for a rectangular prism is V = lwh. I believe a rectangular prism is the best figure for my cereal box design because it has a simple and easy-to-assemble shape, making it cost-effective in terms of production. Additionally, the flat faces of a rectangular prism provide ample space for branding and marketing purposes, allowing me to showcase my cereal's design and product information effectively. Overall, a rectangular prism offers a balance of practicality and visual appeal for my cereal box design.

Answer 2

Measure a standard cereal box and calculate its volume and surface area. Show and label your calculations.

2. Calculate the cost of manufacturing a standard cereal box using a rate of $0.05 per square inch. Show and label
your calculations.
3. Design a new cereal box that will hold the same amount of cereal, but will reduce manufacturing costs. Prove
that your design will hold the same amount of cereal but can be manufactured more cheaply by calculating its
volume, surface area and cost and comparing to the standard cereal box. Show and label your calculations.
4. Draw a net of your new design and include all measurements on your diagram.

Answer 3

[AI-generated answer — Bot GPT 3.5]

1. Measuring a standard cereal box:

Let's assume a standard cereal box has dimensions:
Length = 8 inches
Width = 2 inches
Height = 12 inches

Volume:
V = lwh
V = 8 * 2 * 12
V = 192 cubic inches

Surface Area:
SA = 2lw + 2lh + 2wh
SA = 2(8*2) + 2(8*12) + 2(2*12)
SA = 80 square inches

2. Cost of manufacturing the standard cereal box:
Cost = SA * $0.05
Cost = 80 * $0.05
Cost = $4

3. Designing a new cereal box:
Let's say the new cereal box has the following dimensions to hold the same amount of cereal:
Length = 6 inches
Width = 4 inches
Height = 8 inches

Volume:
V = lwh
V = 6 * 4 * 8
V = 192 cubic inches

Surface Area:
SA = 2lw + 2lh + 2wh
SA = 2(6*4) + 2(6*8) + 2(4*8)
SA = 104 square inches

Cost of manufacturing the new cereal box:
Cost = SA * $0.05
Cost = 104 * $0.05
Cost = $5.20

4. Drawing a net of the new design:
(Not able to draw a diagram)

Answer 4

Write a full paragraph explaining why you think cereal is currently packaged in rectangular prisms. Write an additional

full paragraph discussing the advantages and disadvantages to switching to your new design.

Answer 5

[AI-generated answer — Bot GPT 3.5]

Cereal is currently packaged in rectangular prisms primarily due to their efficiency in terms of storage and transportation. Rectangular prisms are easy to stack and arrange on store shelves and pallets, maximizing space utilization and making it easier to transport large quantities of cereal boxes. Additionally, the flat faces of rectangular prisms provide ample surface area for branding and product information, making them ideal for marketing purposes. The standardized shape also allows for easy storage in kitchen cabinets and pantries at home.

Switching to a new design, such as the one proposed in the previous calculations, could offer several advantages and disadvantages. On the one hand, the new design could potentially reduce manufacturing costs due to its smaller surface area compared to a rectangular prism. This could lead to cost savings for cereal manufacturers in the long run. Additionally, the unique shape of the new design could help the cereal stand out on store shelves, potentially attracting more consumers. However, there could also be disadvantages to switching to a new design. For instance, the new shape may not stack as efficiently as rectangular prisms, leading to increased storage and transportation costs. Additionally, consumers may be resistant to change and may have difficulty storing the new-shaped cereal boxes in their homes. Overall, while there are potential cost and marketing benefits to switching to a new design, careful consideration should be given to the practicalities and potential drawbacks of the change.