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Science, Technology, Engineering and Mathematics
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Which of these infinitely deep vessels will eventually full up?
Why do this problem?
is an exploration into volumes of revolution. The numbers involved are awkward, which will encourage accurate and careful algebra. By thinking about the volumes and the formula students will gain an insight into the types of functions for which a volume of revolution calculation is possible.
This question involves 'awkward' numbers, so might be a good problem to use when practising careful integration. Note that the 'usual' volume of revolution formula is for revolution about the $x$ axis, whereas this problem involves revolution about the $y$ axis. Be sure that students are aware of this problem, which might be overcome in a group discussion.
What is the problem with implementing the usual volume of revolution formula?
Can students explicitly invent a curve which, when revolved, gives a depth of $1$cm when the vessel is half full?
This question is computationally involved. If your students are struggling with the numbers and algebra, focus on the first two curves. You might also change the question so that the curves pass through $(1,1)$ instead of $(2.5,10)$.
Maths Supporting SET
Volumes of revolution
Manipulating algebraic expressions/formulae
Meet the team
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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