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Volume of a smartie
By Jacqueline Chase on January 29, 1998:

Hi. My pupils have been set a project to design a new smarties packet.

Some of them have got hooked up on finding the volume of a smartie. They are GCSE students and have never come across calculus and I have suggested they consider using the volume of either a cylinder or a cuboid as a starting point for them to estimate the volume.

Can anyone think of a way of improving the `model' without getting into the realms of calculus?

Many thanks in anticipation

Jacky C

By Anonymous:

Dear Jacky,

One way would be to think of the smartie as a squashed sphere.

It probably isn't really (I think it is too sharply curved at the edge), but this would give a reasonable approximation to the effect of the curved surface. Using the formula for the volume of a sphere, you should just be able to scale this by the ratio of (the radius of the 'sphere') to (the height of the smartie).

The radius of the sphere here will just be the radius of the smartie.

To find the radius and height of the smartie, it would probably be a good idea to cut the smartie, since then there is a flat face to measure. If your school has a sixth form, the physics department may have a micrometer, which can measure small distances very accurately, but an accurate ruler will probably suffice, since we're already making quite a big approximation.

Alternatively, this might be quite a good way of introducing some of the ideas of calculus. Cut the smartie in half horizontally.

The total volume (V, say) is then twice the volume of each half (V1). But this is just 2×p×radius×(1/2 × area of cross-section). The cross-section is now just a graph of y against x, so would be amenable to say the trapezium rule, which would be accessible (although lots of messy calculations) to GCSE students (I think?) Maybe you could use a computer? To find the shape of the cross-section, you could cut the smartie in half, VERTICALLY this time, and trace round the outline onto a piece of paper, then maybe enlarge the resulting shape using a photocopier. You could then introduce the idea of taking more and more trapezia, and limits etc. Or just counting squares would give quite a good answer.

I hope this has been helpful,

Yours,

David.