### Why do this problem?

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.

### Possible approach

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.

### Key questions

What is the problem with implementing the usual volume of
revolution formula?

### Possible extension

Can students explicitly invent a curve which, when revolved,
gives a depth of $1$cm when the vessel is half full?

### Possible support

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 $(2.5,10)$.