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The 8.5" x 11" paper rectangle has an area of 93.5 in^{2}, but what about its perimeter?
Consider the family of rectangles that have, not a constant area, but a constant perimeter, and answer these questions:
 Make at least 8 physical models of "baseless" cylinders from rectangles with perimeters of 39". You may cut the physical models from paper, draw twodimensional versions (as in our drawing above), or choose some other method.
 For the rectangles with constant area, there were no limits on the height of the cylinder: in theory, the cylinders could become infinitely tall and infinitely narrow, or infinitely short and infinitely stout. Is this also true of the family of cylinders with fixed perimeter? Explain.
 Remember your observation about the cylinders with a constant area: as the cylinders get taller and narrower they hold less, and as they get shorter and stouter they hold more. Through the following exploration, decide whether this is true for constant perimeter as well. (You may need the formula P = 2l+2w for the perimeter of a rectangle.)
Fill in the following chart and compare values with those obtained in the constant area problem. Record your observations.
Write Volume as a function of r. What kind of function do you get?
Graph the function using a graphing utility of your choice. Look at the shape of the graph. Relate the graph's shape to the question we are exploring.
Write a conclusion about the relation between the dimensions of the rectangle and the volume of the cylinder. Is our earlier statement that "as the cylinders get taller and narrower they hold less, and as they get shorter and stouter they hold more" also true for this function? If it is, explain why. If it is not, write a new statement about how volume changes as the dimensions of the cylinder change.
 I discovered an interesting fact about this problem. I'd like you to discover something that you find very interesting. (I am being intentionally vague here because I don't want to stifle your investigation.) You may find my very interesting fact, or you may find something else. Describe your interesting fact.
 Finally, for calculus students only: find the dimensions of the rectangle (w and h) that would yield the greatest cylinder volume given any perimeter p. Explain how to solve this problem using derivatives.
(High School)
(Calculus)
