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## 'Immersion' printed from http://nrich.maths.org/

### Why do this problem?

This problem provides an opportunity to think about what the shape
of a graph means, by relating the graph to the situation which
created it. Students need to reason which graph matches with which
solid, and clearly communicate their ideas. Along the way, there is
the chance to revisit formulas for working out the volumes of
solids.

### Possible approach

Start by asking students to sketch the four solids described in the
problem:

1) A sphere of radius $1$cm

2) A solid cylinder with height $\frac{4}{3}$cm and radius
$1$cm

3) A solid circular cone with base radius $1$cm and height
$4$cm.

4) A solid cylinder of height $\frac{4}{9}$cm with a hole drilled
through it, leaving an annular (ring-shaped) cross-section with
internal radius $1$cm and external radius $2$cm.

Then for each of the solids, students could work out the volume -
this is a good opportunity to discuss the advantages of leaving
answers in terms of $\pi$.

Now hand out

this
worksheet with the five graphs on. Challenge students to label
the axes and the five graphs, annotating each curve along the way
to show key points where interesting things are happening, and how
they relate to the shape of the solid.

Finally, bring the class together to discuss their answers and how
they pieced together which graph is which.

### Key questions

What does a straight line section of graph tell you about the
cross section of the solid being immersed?

What about a curve which gets steeper and steeper? Or one
which flattens off gradually?

Are there any symmetrical features of the solids which help to
identify the correct lines?

### Possible extension

See extension task at the end of the problem.

### Possible support

Maths Filler
offers an opportunity to work with graphs showing changing volumes
based on simple surface areas and graphs made up of straight lines
rather than curves.