CCSS.MATH.CONTENT.7.EE.B.4: Decorative Cake Box

In this activity, you work in a cloud kitchen, a business that has no dining room — only online orders. Cloud kitchens combine pre-planning with just-in-time preparation to optimize kitchen resources and stay competitive.

As the kitchen shuts down for the business day, you are tasked with the responsibility of storing a slice of special order decorative cake in a 9.6 cup plastic storage container. The cake will not be ready until tomorrow, but you already have the dimensions from the customer order.

The cake is 7″ long and 5″ wide , which is no problem because the plastic storage container has the dimensions of 9.5″ x 6.25″. However, the pastry chef is concerned about the height available when the container is sealed with its lid.

The plastic lid is designed to seal the container and protect the freshness of its contents. However, if the decorative cake is somehow damaged, it will become undeliverable and will be discarded as a loss.

The cross-sectional area of the container is a rounded rectangle with a 1″ radius on each corner. We now have a description of the geometric shapes involved, so building a solution requires us to put the pieces together into a system of equations.

CCSS.MATH.CONTENT.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

We begin with a review of our prior knowledge and best practices. First, we want to use inches to communicate the usable height. To be consistent, the units we will use for the volume of the container are cubic inches.

In Wolfram Alpha, type this:
9.6 cups in cubic inches (view query)

For voice assistants, try this command:
9.6 cups in cubic inches

We find that volume of the container, which is 9.6 cups, can be converted to 138.6 cubic inches.

Second, we recall the formula for the volume of a prism:

with base area A for the rounded rectangle and height h. The actual area of a rounded rectangle is defined by:

The variables a and b describe the length and width between the rounded regions of the container, with radius r. We can subtract the radii of the corners from the overall width and length of the container.

We now have enough information to solve the problem.

In Wolfram Alpha, type this:
V=138.6;r=1;L=9.5;W=6.25;a=L-2r;b=W-2r;V=(ab+2r(a+b)+πr^2)h
(view query)

We find that as long as the decorative cake is less than 2.37 inches, the lid should close without damage.

Even though we used equalities in this math practice activity, the concern for product quality may have changed the nature of our expressions into inequalities. For example, notice how we compared the dimensions of the cake slice to the dimensions of the container before any other calculation were made.

Putting in extra work:
Try experimenting with the above calculation in Wolfram Alpha with inequalities. What happens?

Additionally, we may want a factor of safety to further protect the decorative cake, so we may decide eventually that all cakes measuring no more than 2.25 inches will be stored in that particular container. Of course, final decision is made by the pastry chef.

Comparison Shopping:
Try shopping for this and similar plastic storage containers on Amazon. What did you find?

MathForWork delivers distributed learning systems, instruction support, and test preparation for all learners. Learn more at MATHFORWORK.COM. MathForWork is a Bitwise Thermodynamics project.

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