The following activity is an opportunity for math practice in situations when measurement requires math-based design solutions. If possible, read the problem statement aloud. After reading the problem statement, we will have a brief discussion.

## Problem Statement

An empty coffee jar has been converted into a flower vase. The 3D modeling of this coffee jar flower vase is described as two frusta stacked end-to-end to form an hourglass container. The top areas of both frusta share a common square cross-section, which forms a bottleneck at 3.5″ above the bottom of the jar (sitting on a table). The capacity of the container is 800 mL of water. The total height of the container is 5.5″. The base area for the lower frustum and upper frustum are 3.5″ x 3.5″ and 3″ x 3″, respectively. What are the dimensions of the cross-sectional area at the bottleneck?

## Discussion

Wow! 😂😂😂 Lots of words. That is the point.

This activity is designed to help you practice reading a lot of words in test items. You will gain experience organizing these words and ideas using computational thinking.

Let us take a moment to visualize the solid model described above with 3D file viewer on GitHub. Or you can simply view the video below:

Before we begin this problem, we need to recall some prior knowledge and best practices. **First, we choose and confirm the units of the final answer.** In this case, the units are [inches x inches] or [in. x in.].

Note that the question did not ask for the side length or the actual area. The call to action asked for the * dimensions*. In the workplace, it is very important to pay attention to what the request for proposal (RFP) needs.

**Second, we convert all quantities into compatible units.** We convert 800 millimeters into cubic inches. The result is about 49 cubic inches.

In Wolfram Alpha, type this:`800 ml in cu in`

(view)

**For voice assistants, try this command: 800 milliliters in cubic inches**

**CCSS.MATH.CONTENT.5.MD.A.1: **Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Next, we look at the formula for the **volume of a frustum. **

There are two frusta in this problem, so we will be using the same formula twice to help solve for a system of equations.

**Let us place the container on the table.** The table will be our reference point. The total height of the container is the distance from the table to the top of the container (see **Figure 1**). The total height (in inches) of the container is *H = 5.5*.

This measurement excludes the the jar mouth section, where the screw top lid has been removed.

For the *lower frustum volume Y*, we suppose that the base area (in inches squared) is *A = 3.5 * 3.5* and the height (in inches) from the table is *h = 3.5*.

The units of area are square inches. We will let Wolfram Alpha make the calculations for us by leaving the dimensions as multiplicands. This methods lets us make design changes to the dimensions as needed without re-building our solution.

For the *upper frustum volume Z*, we conclude that the base area is *B = 3 * 3*, with a height of *H – h*.

We know the exact height of the upper frustum, but we will still use variables to express the difference between the total height and the height from the table.

Both frusta share the common bottleneck area *X*. Their combined volume is 49 cubic inches; that is, *Y + Z = 49.*

Our analysis is complete. We now know enough information to solve for the unknown bottleneck area X.

In Wolfram Alpha, type this:`Y + Z = 49; H = 5.5; A = 3.5 * 3.5; B = 3 * 3; h = 3.5; Y = (h/3) * (A + X + SQRT(A*X)); Z = ((H-h)/3) * (B + X + SQRT(B*X)); J = SQRT(X);`

(view)

The dimensions of the bottleneck cross-section are **2.63″ x 2.63″.**

**Key vocabulary:** request for proposal, units, unit conversion

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