
You're a part of a marketing firm that is studying the packaging of soft drinks. You've seen soft drinks packaged in sixes. How efficiently does this arrangement use shelf space? Do other arrangements fit more cans on a shelf? In this activity you will explore these ideas. For starters, consider six cans packed in rows in a rectangular package, the standard six-pack. The following figure shows a two-dimensional view of it, with each circle representing the bottom of a can. Use the following questions to help you determine how efficient the package is. Be sure to justify your answers and explain how you found them. The efficiency of a package is the ratio of the total area of the bottoms of the cans to the total area of the bottom of the box. 1. What are the dimensions of the bottom of the box if the can radius is 3 cm? __________________ 2. Why is this definition reasonable? 3. What is the efficiency of the standard six-pack as pictured above? State your answer as a fraction in lowest terms and leave p in your fraction. _______________________________ 4. Express the efficiency as a percent rounded to three decimal places. ___________________ 5. Calculate the efficiency for a square-based one-pack. _____________ Is it the same or different from the six-pack? Explain what you found and why it turned out that way. 6. What would you expect the efficiency to be for a square-based four-pack? Calculate and check your prediction. Can you generalize to any pack that consists of a combination of square-based one-packs? 7. Calculate the efficiency of a standard, rectangular six-pack of cans if the radius is a) 4 cm efficiency= ______% b) 5 cm efficiency= ______% c) 6cm efficiency= _______% Explain what you notice! 8. Use an algebraic argument to determine what happens in problem 7 by calculating the efficiency of the standard six-pack if the radius of the can is r cm.

So far you have been looking at rectangular arrangements. What happens if you consider other shapes for the bottoms of the boxes? The focus of this activity is to examine parallelograms. Assume that the radius is r cm. Explain your calculations. 9. What is the area of the parallelogram? ___________ (Hint: The following figure shows many radii. Use it to show that the little triangle is equilateral and draw some conclusions about the angles in the figure. Then use what you know about 30-60-90 triangles or the Pythagorean theorem to find the base and height of the parallelogram.) 10. What is the efficiency of the parallelo-pack? (Show your work below. Express it as a fraction and a percent.) _______________ 11. How does the efficiency of the parallelo-pack compare with that of the rectangular six-pack? The rectangular six-pack arranged the cans in a stack position, whereas the parallelo-pack uses a staggered arrangement. 12. Discuss what effect you think the stack and stagger methods might have on the efficiency of each shape of package. 13. What other aspects about packaging cans using the parallelo-pack might manufacturers, wholesalers, distributors, retailers, and consumers be interested in?

14. Find the efficiency of each of the following containers: (sketch each and show your calculations on a separate piece of paper. Record your final drawings and efficiencies on this page - very neatly!) THREE - Packs
a) Isosceles right triangle	b) Equilateral triangle
FIVE - Packs
c) Equilateral Triangle on a Square	d) Isosceles Trapezoid (half of a regular hexagon)
SEVEN - Packs
f) Circle	g) regular hexagon
The design of soft-drink containers involves many other considerations besides efficiency as we have defined it. One particular factor to examine is how the package fits in its environment - shelf efficiency. Can the boxes be placed next to one another without wasting space? The process of filling a plane with a particular shape is called tessellating the plane. 15. Which of the designs above will tessellate the plane? If the shape will tessellate the plane, trace each shape and make a model that you can use to show how well the shape fills the plane. Why do you think the rectangle was the shape of choice for the standard six-pack even though you probably have already discovered that it is not the most efficient? Is the rectangle the only shape that tessellates the plane? On to the Popzi Challenge