Susan BooneSaint Agnes Academy, Houston, Texas
Susan Boone
Saint Agnes Academy, Houston, Texas
September 1999
written in conjunction with TERC
and the The Math Forum
through
BRAP--Bridging
Research and Practice.
Objective:
Students will work with given formulas to solve for specific variables. They will calculate the volume of a cylinder and analyze the change in volume when there is a constant surface area. They will repeat the problem with a constant perimeter.
Materials:
- 8 1/2 by 11 sheets of paper for the class (transparencies work well for the initial experiment)
- larger sheets of paper to make cylinders in other sizes (butcher paper works well, but cut it prior to class)
- tape, ruler, graph paper, fill material (Rice Krispies, Cheerios, packing "peanuts", etc.)
- formulas for circumference of a circle: 2 Pi r, and volume of a cylinder:
Pi r2 h
- graphing calculator or spreadsheet for computer
Procedure:
Demonstrate how to make a cylinder from a regular sheet of paper. Note that there is no top or bottom in this model.
Problem:The paper represents the lateral surface area. Ask for suggestions for making cylinders in other sizes, and model these suggestions. Focus on the two cylinders that could be made by rolling the rectangle two ways, so that the height is either 11" or 8 1/2". Ask students how the volumes compare. Make these two cylinders using a transparency (colored ones are nice; I have also used halves of report folders). Tape the edges together with no overlap.
Place the 11" cylinder in a large flat box. Use the fill material to fill the cylinder that is 11" high. Place the 8 1/2" cylinder around it and ask for predictions for how full the outer cylinder will be if the 11" cylinder is lifted out so that the fill material gods into the outer cylinder. Students should record their speculations prior to actually doing this experiment.
I like to set this problem up as a scientific experiment. In their writeups, students should include a hypothesis, data, results, and a conclusion.
Consider the family of cylinders that could be made having a fixed area of 93.5 (the area of an 8 1/2" by 11" sheet of paper). What are the dimensions of the rectangle (to the nearest 1/2") that would produce the cylinder with the greatest volume?
(Optional)* Repeat this procedure with rectangles that have a fixed perimeter of 39" (the perimeter of an 8 1/2" by 11" sheet of paper). What are the dimensions of the rectangle (to the nearest 1/2") that would produce the cylinder with the greatest volume?
Hypotheses:
Students should predict the sizes of the rectangles that would produce the cylinder with the largest volume.
Data:
Write an equation for the volume of the cylinder as a function of its radius. (I provide formulas for circumference and volume of a cylinder.) Enter these data into your graphing calculator. Make a table (using your graphing calculator) that models the volume of cylinders that have a constant surface area. Record these results.
Repeat this procedure, calculating the volume as a function of its height.
Results:
What are the dimensions of the cylinder that gives the largest volume? Is that the largest volume possible? Graph the data: x = radius, y = volume. Explain the graph. Draw another graph with x = height, y = volume.
Conclusion:
Write a conclusion that explains the data you collected and the graphs you drew. Is there a maximum volume for both situations? If there is, what is it? Why do you think the greatest volume was achieved with these dimensions? If there was not, why not? Explain.
These pages were developed through GirlTECH , a teacher training and student technology council program sponsored by the Center for Research on Parallel Computation (CRPC), a National Science Foundation Science and Technology Center.
Copyright February 2000 by Susan BooneThanks to the RGK Foundation for its generous support of GirlTECH.