
Objective: Students will build a family of cylinders and discover the relation between the dimensions of the generating rectangle and the resulting pair of cylinders. They will order the cylinders by their volumes and draw a conclusion about the relation between a cylinder's dimensions and its volume. They will also calculate the volumes of the family of cylinders with constant area. Finally, they will write the volumes of the cylinders as a function of radius and, using derivatives, find the cylinder with the greatest volume, given a fixed perimeter. Materials: 8 1/2" by 11" sheets of paper for the class (transparencies work well for the initial experiment), tape, ruler, graph paper, fill material (birdseed, Rice Krispies, Cheerios, packing "peanuts," etc.). Vocabulary: Cylinder, dimension, area, circumference, height, lateral surface area, volume Procedure: Initial Experiment: Take a sheet of paper and join the top and bottom edges to form a cylinder. The edges should meet exactly, with no gaps or overlap. With another sheet of paper the same size and aligned the same way, join the left and right edges to make another cylinder.
Stand both cylinders on a table. One of the cylinders will be tall and narrow; the other will be short and stout. We will refer to the tall cylinder as cylinder A and the short one as cylinder B. Mark each cylinder now to avoid confusion later. Mark each cylinder now to avoid confusion later. Now pose the following question to the class: "Do you think the two cylinders will hold the same amount? Or will one hold more than the other? If you think that one will hold more, which one will that be?" Have them record their predictions, with an explanation. Place cylinder B in a large flat box with cylinder A inside it. Fill cylinder A. Ask for someone to restate his or her predictions and explanation. With flair, slowly lift cylinder A so that the filler material falls into cylinder B. (You might want to pause partway through, to allow them to think about their answers.) Since the filler material does not fill cylinder B, we can conclude that cylinder B holds more than cylinder A. Ask the class: "Was your prediction correct? Do the two cylinders hold the same amount? Why or why not? Can we explain why they don't?" (Note to the teacher: because the volume of the cylinder equals pi*r^{2}*h, r has more effect than h [because r is squared], and therefore the cylinder with the greater radius will have the greater volume.) Second Experiment: "Let's go back and look at our original sheet of paper. We made two different cylinders from it. What geometric shape is the sheet of paper?" (rectangle) "What are its dimensions?" (8.5" by 11"). "What are the dimensions of the resulting cylinders? That is, what is the height and what is the circumference?" (The height of the cylinder is the length of the side of the paper rectangle that you taped, and the circumference is the length of the other side.) "Are there any other cylinders that we can make from this same sheet of paper?" (Yes. There are many cylinders that can be made.) "Let's try to make some other cylinders. If we fold a new sheet of paper lengthwise and cut it in half, we will get two pieces  each measuring 4.25" by 11"  which we can tape together to form a rectangle 4.25" by 22". We can repeat the process to create a second rectangle the same size. Now we can roll these rectangles into two different cylinders, one 4.25" high and another 22" high. We will label them cylinder C (4.25" high) and cylinder D (22" high)." "Now we have four cylinders. Which of them would hold the most? Write down your predictions."
Test by filling. Have a student report the results. Now have the students arrange the cylinders in order, by volume, from the cylinder that holds the least to the cylinder that holds the most. "Do you see any pattern that relates the size of the cylinder and the amounts they hold?" (As they get taller and narrower the cylinders hold less, and as they get shorter and stouter, they hold more.) "How many other cylinders could we make from a rectangle with these same dimensions?" (Theoretically, infinitely many. Cylinders could get taller and narrower and taller and narrower until they were infinitely tall and infinitely narrow, or they could get shorter and stouter and shorter and stouter until they were infinitely short and infinitely stout.) Calculation Investigation: "We think that the taller the cylinder, the smaller the volume, and the shorter the cylinder, the greater the volume. Can we write this in mathematical language that will help us confirm our observations? What formulas relate to this problem?" C = 2pi*r or pi*d [circumference of a circle]So if our ultimate goal is to calculate the volume, then the formula we will need to use is Find h: "Let's go back to our original sheet of paper. What were its dimensions?" (8.5" by 11") "Which of these two dimensions represents the height of the cylinder?" (11". The height of the taped edge of the paper is the height of the cylinder.) "Half way there. We have found h. Now on to r." Find r: "How does the circumference of the cylinder relate to the dimensions of the rectangle?" (The base of the rectangle is the circumference of the cylinder.) "So, since the circumference is equal to 2pi*r and the circumference equals the base of the rectangle, then C = 2pi*r = 8.5" So now we can solve for r. How do we do that?" (Divide both sides of the equation by 2pi.) "What do we get?" (r = 8.5/(2pi) = 1.35282") "Now we have r and h and we are ready to find the volume. Let's put them both into the volume formula, V = pi*r^{2}*h Using substitution,"Now you do the other cylinder and see what you get. Compare the volumes of the two cylinders. Do your results confirm what we discovered with our physical models?" (Note to teacher: you may need to lead students through the reasoning here as well.)
Organizing Material: Complete a Table Remember our conclusion relating the dimensions of the cylinder to its volume? (As cylinders get taller and narrower they hold less, and as they get shorter and stouter they hold more.) Fill out the following table, and confirm that calculation. You can download the completed table as an Excel spreadsheet here.
Extension Question: Multiply the number in the first column of the above table by the number in the second column. What do you notice? (The products are all equal.) Why is this true? (These products represent the base times the height of the rectangle  in other words, the area. Since the cylinders were all made from sheets of paper having the same dimensions, they all have the same area. The rectangle area represents the lateral surface area in the cylinder.) Assessment Question: Find a cylinder that has a volume of over 300 in^{3} with a lateral surface area of 93.5 in^{2}. In the above table, fill in all four columns for that cylinder. Using Algebra: Consider the whole family of cylinders that you could make with a fixed lateral surface area of 93.5 in^{2}. Remember how many there would be? (Theoretically, infinitely many.) Write an expression for the volume of the cylinders as a function of r.
(Note to the teacher: Students need to learn to see the relation between r and h, so they can rewrite h in terms of r in the formula V = pi*r^{2}[93.5/(2pi*r)]What kind of function do you get? (linear) How could you use figures from the table to check your equation? (Pick an r from the table, put it into the equation, and see whether you get the correct V. [How do I know I'm right?]) Assessment Questions: Using our function, find the volume when r = 50, or when r = .005. Does the function confirm our earlier observation that as the cylinders get taller and narrower they hold less, and as they get shorter and stouter, they hold more? Explain. Note to the teacher: you may want to assign the Constant Perimeter Project as an outofclass or a nextday project, depending on the amount of guidance you feel you will need to give. Assessment: Evaluate answers from group projects. If you want to do further evaluation, you can add a similar problem on your next test using a differentsized sheet of paper. Example: Find the height of a "baseless" cylinder that would yield the maximum volume given a family of cylinders made from rectangles with a constant perimeter of 40. Extension: After students have completed the Constant Perimeter Project, talk about the concept of maximum volume. Notice that the rectangle that gives the greatest volume is twice as wide as it is high. In other words, p/6 is the height, and p/3 is the circumference that gives the greatest volume. This is the interesting fact described in the activity that follows. Have students explore whether this is a unique occurrence that applies only to this particular perimeter, or whether it seems to be true given any P.

