VISUALIZING THE METHOD OF FINDING VOLUMES BY CROSS SECTIONS: AN EGGSPERIMENT

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Jun 2006 by Uhl, Jean, Humphrey, Patricia, Braselton, James

ABSTRACT:

For some students, using the method of cross sections to find the volume of a solid is difficult to understand because of the abstraction involved. This paper deomonstrates a fun way for students to visualize the method of cross sections. Although a fun activity, the paper implicitly conveys the importance of connecting theory and experiment.

KEYWORDS: Calculus, model, volume, cross sections, mean, standard deviation, teamwork.

1 INTRODUCTION

For the student who cannot visualize the solid, the confusion lies in determining "where are the cross sections?" A thoughtful instructor will usually try to convince the student that the cross sections are coming out of the blackboard or will even make an attempt at a free-form three-dimensional drawing to convey the concept.

While teaching this concept recently, we developed a real-life analogy (possibly motivated by the fact that one of us had just made a salad involving sliced eggs) using hard-cooked eggs. (Refer to any standard cookbook like The Joy of Cooking by Rombauer and Becker, [2], for more details regarding egg preparation techniques.) The students and instructors had a lot of fun with the experiment and we believe that the students might have a better intuitive idea of the method of finding volumes of a solid by cross sections than others. Although we did not explicitly state so to our students, we tried to convey the interplay between theory and scientific evidence. An interdisciplinary approach (engineering, mathematics, statistics) that involved a collaborative teaching effort made the experience special for us and meaningful to our students.

We calculated the volume of an egg in two ways. First, we used calculus to develop a formula to approximate the volume of an egg based on actual egg measurements. This number was then compared to the volume of water displaced by the same egg. Although these numbers varied, our statistical analysis indicates that the calculus formula we derived generally provides a decent approximation of the volume of a good egg.

2 VOLUME OF AN ELLIPSOID

If you need to motivate the eggsperimeni, peel one hard-cooked egg and then use your egg slicer to slice the egg perpendicular to its major axis. (See Section 3 for a list of supplies needed for the eggsperiment.) Show your class that the cross-sections are circles.

The egg is not an exact ellipsoid: going left to right above the real egg is not symmetric with respect to the vertical axis, but it is close. We do a half revolution to illustrate the method of finding the volume of a solid with known base and cross-section so we approximate the egg's volume by finding the length of its major and minor axes and view the egg as a solid with cross-sections of known area as illustrated in Figure 2. On the other hand, if you wanted to illustrate the method of disks, adjust the equations accordingly and do a full revolution.

3 THE EGGSPERIMENT

To carry out the eggspenment, you will need

1. Vernier calipers (at least one per group).

2. One or more graduated cylinders or measuring cups (groups can share).

3. Egg slicer (optional).

4. Hard-cooked eggs. Students can work in groups of two or three. Depending on your class size 7 to 15 eggs should be sufficient to make your point. Bring along an extra hard-cooked egg if you wish to use an egg slicer to show your class that cross sections perpendicular to the major axis of the egg are circles. As the eggs will be handled by several individuals, we recommend that you use unshelled eggs to avoid breakage and for sanitary considerations, especially if the eggs are made into sandwiches afterwards.

5. Calculator with basic statistical capabilities, access to a statistical software package, or access to a computer algebra system.

6. Salt, pepper, mayonnaise, knife, and bread (all optional).

Each student team is given an unshelled egg1 and required to measure the width and height three times. A quick lesson in reading the scale on the vernier may be necessary to ensure proper readings. If the measurements are taken in centimeters, this will facilitate comparison to milliliters easily because 1 cm^sup 3^ is equal to 1 milliliter. After the three height and width measurements are taken, they are then averaged to maintain the best accuracy possible. Using the averaged width and height measurement, the calculations of formula 3 are carried out for the individual egg. Next, the graduated cylinder or measuring cup is filled about half way with water and the level noted. Starting at a round number, such as 250 milliliters, is recommended. The two team members can verify the initial height and record the number. Next, the whole egg is placed into the water and the new level recorded. The accuracy of the reading depends on the graduations of the cylinder or the measuring cup. It's best to have the whole team read the volume displaced and agree on the measurement. Subtract the initially recorded level from the new level from and this is the volume of the egg using the displacement test. This volume is in milliliters and can be directly compared to the volume calculated using formula 3.