Your goal is to design a new shape for a cereal container that will hold the same volume of cereal as a standard cereal box and costs as little as possible to make. Assume that cardboard costs $0.05 per square inch. How cheaply can you package cereal
One possible design for a new cereal container that would hold the same volume as a standard cereal box and cost as little as possible to make is a cylindrical container.
First, let's determine the volume of a standard cereal box. A standard cereal box typically has dimensions of around 3 inches x 10 inches x 12 inches, resulting in a volume of 360 cubic inches.
Next, let's calculate the surface area of a cylinder that would hold the same volume. We can use the formula for the volume of a cylinder, which is V = πr^2h, where r is the radius and h is the height. Since we want the same volume as the standard cereal box, we can set V equal to 360 cubic inches.
Let's assume the height of the cylinder is 10 inches (same as the standard cereal box) to simplify calculations. By rearranging the formula for the volume of a cylinder, we get r = sqrt(360 / (10π)) ≈ 3.015 inches.
Using the surface area formula for a cylinder, which is A = 2πrh + 2πr^2, we can calculate the surface area of the cylinder. The surface area comes out to be approximately 190.46 square inches.
Finally, by multiplying the surface area by the cost per square inch ($0.05), we can determine the cost to make the cylindrical cereal container. The total cost would be $9.52.
Thus, by designing a cylindrical container for the cereal, we can package the same volume of cereal for $9.52, costing as little as possible to make.
You will turn in the results of this activity for your assessment for this lesson.
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.
Volume of a standard cereal box:
Dimensions of a standard cereal box: 3 inches x 10 inches x 12 inches
Volume = Length x Width x Height = 3 in x 10 in x 12 in = 360 cubic inches
Surface area of a standard cereal box:
Surface area = 2lw + 2lh + 2wh = 2(3 in x 10 in) + 2(3 in x 12 in) + 2(10 in x 12 in) = 60 + 72 + 240 = 372 square inches
Cost of manufacturing a standard cereal box:
Cost = Surface area x Cost per square inch = 372 sq in x $0.05 = $18.60
New design for a cereal box:
Considering a cylindrical container design.
Given:
Volume = 360 cubic inches
Height = 10 inches (same as standard cereal box)
Radius, r = sqrt(360 / (10π)) ≈ 3.015 inches
Surface area of the new cylindrical cereal container:
Surface area = 2πrh + 2πr^2 = 2π(3.015 in)(10 in) + 2π(3.015 in)^2
Surface area ≈ 190.46 square inches
Cost of manufacturing the new cylindrical cereal container:
Cost = Surface area x Cost per square inch = 190.46 sq in x $0.05 = $9.52
Conclusion:
The new cylindrical cereal container holds the same volume as a standard cereal box but can be manufactured more cheaply. The cost to manufacture the new design is $9.52, which is more cost-effective than the $18.60 required for the standard cereal box.
Net of the cylindrical cereal container:
Below is the net of the new cylindrical cereal container design:
```
________
/ /\
/_____/ \
\ \ \
\_____\____\
Height = 10 inches
Radius = 3.015 inches
```
This net design can be used for mass-producing the new cost-effective cylindrical cereal containers.
You will turn in the following:
a table containing the measurements of a standard cereal box
clearly labeled work showing how you calculated the volume of a standard cereal box
clearly labeled work showing how you calculated the surface area of a standard cereal box
clearly labeled work showing how you found the cost of producing a standard cereal box
an explanation proving that your new design holds the same amount of cereal as a standard cereal box but that it costs less to manufacture
a net of your new design, including any measurements that would be necessary to produce the cereal box
I will provide the information requested:
Table containing the measurements of a standard cereal box:
| Dimension | Length (in) | Width (in) | Height (in) |
|-----------|-------------|------------|-------------|
| Measurement| 3 | 10 | 12 |
Calculation of the volume of a standard cereal box:
Standard cereal box volume = Length x Width x Height
Standard cereal box volume = 3 in x 10 in x 12 in
Standard cereal box volume = 360 cubic inches
Calculation of the surface area of a standard cereal box:
Standard cereal box Surface Area = 2lw + 2lh + 2wh
Standard cereal box Surface Area = 2(3 in x 10 in) + 2(3 in x 12 in) + 2(10 in x 12 in)
Standard cereal box Surface Area = 60 + 72 + 240
Standard cereal box Surface Area = 372 square inches
Calculation of the cost of producing a standard cereal box:
Cost of producing a standard cereal box = Surface Area x Cost per square inch
Cost of producing a standard cereal box = 372 sq in x $0.05
Cost of producing a standard cereal box = $18.60
Explanation of cost savings with the new design:
The new cylindrical cereal container design has the same volume as the standard cereal box. The surface area of the new cylindrical container is 190.46 sq in compared to the standard box's 372 sq in. This reduction in surface area allows for a cost of $9.52 to manufacture the cylindrical container, which is less expensive than the $18.60 required for the standard cereal box.
Net of the new cylindrical cereal container:
```
________
/ /\
/_____/ \
\ \ \
\_____\____\
Height = 10 inches
Radius = 3.015 inches
```
The above net design includes the relevant measurements needed for the mass production of the new cost-effective cylindrical cereal containers.
a paragraph explaining why cereal is packaged in rectangular boxes
a paragraph explaining the advantages and disadvantages of your new design
Cereal is typically packaged in rectangular boxes due to practical reasons related to storage, stacking, and display. Rectangular boxes are easy to stack on shelves, store in pantries, and arrange neatly for retail display. The consistent shape of rectangular boxes makes them efficient for shipping and handling, as they can be packed tightly together, reducing wasted space during transportation. Additionally, the rectangular form factor allows for easy pouring and scooping of cereal from the box, making it convenient for consumers to access the product.
The new cylindrical cereal container design presents several advantages and some disadvantages compared to the traditional rectangular box. One of the main advantages is the reduced manufacturing cost, making it a more cost-effective option for cereal companies. The cylindrical shape is also unique and eye-catching, potentially standing out on store shelves and attracting consumers. However, the cylindrical shape may pose challenges in terms of storage and stacking efficiency, as it may not fit as neatly in pantries or on shelves compared to rectangular boxes. Pouring cereal from a cylindrical container may also be less convenient for some consumers, as accessing the cereal from a round opening can be trickier than from a wide rectangular flap. Ultimately, the new design offers cost savings and aesthetic appeal but may require adjustments in handling and storage for both producers and consumers.