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Cylinder inquiry

For a cylinder of volume πr2h, r + h = n. Given n, the maximum volume occurs when r and h are integers.
This inquiry is suitable for older secondary school students who are familiar with the inquiry process. They need resilience to unpick the prompt - to explain the formula (or request a teacher's explanation) and to identify the constraint. Once the prompt is clear, students seem quick to speculate that it must be false because there are "so many types of numbers". However, when n is defined as a whole number, the prompt is true if n is a multiple of three. Students have reached this conclusion through the realisation that the maximum volume occurs when rn.
   
In the first phase of this inquiry, students' questions often far outweigh statements or speculations:
    • What does the prompt mean? 
    • Why is the volume of a cylinder πr2h? Is it always the same? 
    • Can you explain the formula?
    • What are r, h and n
    • Is n a whole number?
    • Does the answer depend on the value of n?
    • Why do r and h have to equal n?
After a lengthy orientation phase, classes have decided to test the prompt by developing a collaborative inquiry with groups or pairs of students exploring for different values of n (restricted to whole numbers at first). The results for the values of r and h that give the maximum volume can be collected together in a table from which students induce the relationship between r, h and n by spotting patterns. Alternatively, students have graphed the volume given by each radius for a particular value of n, paying particular attention to the stationary point when it lies between whole number values of the radius. The class has gone on to compare the graphs, discussing why each one looks similar no matter what the value of n.
  
The inquiry has moved on to allowing n to be a terminating decimal. 
While this has not yielded any significant new insights, students have shown that any decimal divisible by three has a solution with the same number of decimal places as n. So if n = 6.18, for example, the maximum volume is given by r = 4.12 and h = 2.06. This has led students to attempt re-writing the contention in the prompt: "Given n to a specific degree of accuracy, r and h (that give the maximum volume) have the same degree of accuracy when n is divisible by three."
   
Up to this point i
n the inquiry, students have often been heavily involved in calculating the volume of cylinders. It can be necessary, therefore, for the teacher to guide the class towards reflecting on exactly what has happened and what it means in regard to the aim set at the start - that is, to evaluate if the prompt is true. Furthermore, students should be encouraged to consider why the maximum volume occurs when the ratio of radius to height is 2:1.
  
Pathways
While there are a number of pathways to pursue at this point, each one could result in another round of exploratory calculations leading to the induction of a new result. The inquiry teacher might judge this to be too repetitive and 
introduce calculus to prove the general result (see below). By working out a derivative, students can deduce the new results, rather than continue to substitute numbers into a formula.
Proving the maximum volume occurs at rn

Further pathways:
  • Change the constraint in the prompt - for example, 2r + h = n.
  • Find the maximum volume of other solids given a constraint - for example, the volume of a pyramid with the sum of the base side length and height equal to some n.
  • Explore the r and h that give the minimum volume.

Resources 
Prompt sheet
Promethean flipchart   download