Question

A candy company has hired you as their new production manager. Your job is to choose the form of packaging for a new product from the five choices below. The candy is animal gummies. The package must hold between 45 and 50 cubic inches (roughly 3 cups of space). The cost of the plastic for the packaging is $0.002 per square inch. The company would really like the container to be in the shape of Noah’s Ark (for reference, Noah's instructions are given to him by God (Genesis 6:14–16): the ark is to be 300 cubits long, 50 cubits wide and 30 cubits high. 1 cubit = 1.5 feet.). To simplify things, we are assuming the ark is the shape of a rectangular prism. Your project should include the volume, total surface area, and materials cost for each solid given below, including the formulas you used and each step of your work. Make sure to use the formulas given in your lessons, round your volume and area to the nearest hundredth, and round your cost to the nearest penny. You also need to include a paragraph that answers the following questions:

Do all of the solids hold between 45 and 50 cubic inches?
Which solid is the most cost-efficient (the packaging with the smallest materials cost that holds between 45 and 50 cubic inches)?
Would the Ark be the most cost-efficient solid for packaging? Why or why not?
Which solid would you recommend using? Why?

Answer 1

To choose the most cost-efficient packaging for candy in the shape of Noah’s Ark, one must calculate the volume, surface area, and material cost of the potential designs to find the ideal balance between capacity, theme, and cost.

Step-by-step explanation:

Choosing the Cost-Efficient Packaging for Candy

The task is to select the most cost-efficient packaging for animal gummies that resembles Noah’s Ark, ensuring the package holds between 45 and 50 cubic inches of volume. Based on the given dimensions of Noah’s Ark (300 cubits long, 50 cubits wide, and 30 cubits high), the ark-shaped container must be scaled down appropriately to meet the volume requirement.

To calculate the volume (V) of the rectangular prism, we can use the formula V = length × width × height. Similarly, the surface area (SA) is found with the formula SA = 2(length × width) + 2(length × height) + 2(width × height). The cost is determined by multiplying the total surface area by the cost per square inch of plastic. Consequently, we need to find the dimensions of the rectangular prism that not only fit the volume criteria but also make the packaging as cost-efficient as possible. After calculating the volume, SA, and costs for all potential containers, we can decide which one fits the criteria best.

While the biblical proportions of Noah’s Ark are large, its miniaturized form must also be cost-effective and within the capacity range. Among the given options, we must find the one that maximizes material efficiency and cost-effectiveness. Finding the most efficient packaging may involve trading off between a more exact Ark shape and a simpler, less expensive design.

Upon calculation, we can then determine if any of the proposed solids hold the correct volume range and which is the most cost-efficient. The Ark shaped container, while thematically appropriate, may not necessarily be the most cost-efficient due to potentially higher surface area to volume ratio. The recommended solid will be the one that best balances thematic resemblance, capacity requirements, and cost efficiency.