Juice Box Geometry

Objective

In this experiment you will measure the dimensions of different juice box products to find out which manufacturer has the largest volume of juice and uses the least amount of packaging material.

Introduction

Geometry is the study of how to use math to describe and investigate different points, lines and shapes. The way that a shape is described in geometry is with a formula, which is simply a mathematical way to calculate different properties of a shape: length, size, area or volume. Volume is a unique property of three-dimensional (3-D) shapes because three-dimensional shapes take up space in three different directions.

Most real world objects take up space, have a measurable volume, and are three-dimensional: toys, balls, food, cars, etc. You can use formulas to make geometrical models of common household objects that have 3-D rectangular shapes. A rectangular prism is a shape like a cereal box, or any other box for that matter because rectangular prisms happen to make excellent containers.

In this experiment you will use geometry to produce a mathematical model of a juice box. You will measure rectangular prisms (juice boxes) and describe their properties with geometrical formulas for volume (how much space an object fills in 3-D space) and surface area (how much area is on the outer surface of the object). Each of these formulas can be calculated with three basic measurements: length, width and height.

In the case of a juice box, the rectangular container holds a liquid (juice) that has already been measured by the manufacturer in ounces. This is handy because we can check our calculation of the volume of the container to see if it matches the volume of liquid held inside. Also, we can compare how much packaging material we need to make a container big enough to hold the juice. By modeling juice boxes from different brands and manufacturers, we will investigate if some brands use more packaging material than other brands.

Terms, Concepts and Questions to Start Background Research

To do this type of experiment you should know what the following terms mean. Have an adult help you search the Internet, or take you to your local library to find out more!

• three-dimensional (3-D) object
• length (l)
• width (w)
• height (h)
• volume (V)
• surface area (SA)

Bibliography

• Here is the link to download a program for calculating different geometrical formulas that you will use for this experiment:
  East, T., 2005. "Educational Software Index: Geometry." [accessed: 3/3/06] http://homepage.mac.com/teast/index.html
• This site is a really good site for kids to review math skills, solve puzzles and read stories about math. They also have a section on geometry:
  Karen, 2005. "Cool Math 4 Kids," Coolmath.com, Inc. [accessed: 3/3/06] http://www.coolmath4kids.com/
• NCES, 2006. "Create a Graph," National Center for Education Statistics (NCES) U.S. Dept. of Education. [accessed: 3/3/06] http://nces.ed.gov/nceskids/createagraph/

Materials and Equipment

• metric ruler
• different brands of juice boxes
• graph paper
• the Geometry Application by Travis East (see Bibliography)

Experimental Procedure

1. This experiment uses the Geometry Application by Travis East (see Bibliography). Have an adult help you download the application (be sure to choose the correct version for your computer and operating system) and install it on your computer.
2. In this experiment you will be measuring the sides of several juice boxes to calculate the amount of packaging that each juice box is made of and how much juice it holds. You will need a data table to keep track of your measurements and other data:

3. Take the first juice box and write the name of the brand and the volume of juice in ounces (found towards the bottom of the box) into the data table.
4. Using a metric ruler, measure the height, width and length of the juice box in centimeters and write the measurements into the data table.
5. Open the Geometry application, select "3-D Area & Volume," then select "Rectangular Prism."

6. Type in your values for length, width and height into the boxes and click on "Calculate."

7. Write the data for "Volume" and "Surface Area" into your data table. The "Volume" will be in cubic units (cm³) and the "Surface Area" will be in squared units (cm²).
8. Make a bar graph of surface area and volume for each brand of juice box. You can make your graphs by hand or you can try using the Create a Graph web site for kids from the National Center for Education Statistics.
9. Which brands use the least amount of packaging material? Compare the fluid ounces to calculated volume, how do they compare? Are they the same? Which brands give you the most juice per juice box?

Variations

• This experiment works for containers that are 3-D rectangular prisms, but packaging comes in all shapes and forms. Can you use the principles of geometry to investigate other forms of packaging? Can you make geometrical models for other products? How do cylindrical ice cream tubs compare to rectangular ice cream tubs?
• A more advanced way to compare the different brands is to calculate the ratio of Surface Area to Volume. Sometimes the ratio is written as "SA:V" and sometimes it is expressed as a fraction, "SA/V." Either way, compute this calculation by dividing the SA by the Volume. The closer to one your answer is, the more equal the ratio is. The higher the number, the more packaging is used per volume of juice. The lower the number, the less packaging is used per volume of juice. Compare the different ratios, which brands maximized the SA:V ratio to reduce packaging costs?
• In this experiment you are using containers that are basically the same size and that hold juice. What about other packages for other products? How can geometry help reduce waste in packaging material? Is there less packaging waste buying a large box of cereal or a small box of cereal? How about the single use cereal boxes? Is it better to buy individually wrapped raisin boxes, or a big box of raisins?

Source: http://www.sciencebuddies.org/mentoring/project_ideas/home_Math.shtml?from=Home

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