Stephen ARNOLD
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## Half a Glass Please!

At a restaurant recently, the waiter offered to fill my glass.

Since I was driving, I asked for just half a glass.

To what height should the glass be filled?

Study the model shown above. The sample glass displayed in the GeoGebra app has height 20 cm, with a stem of 10 cm. The rim has diameter 2 cm and narrows to 1 cm diameter (I know - not a very accurate model, but it suits for our purposes here!).

1. Drag the point marked "a" to explore this problem: at what depth is the glass half full?

2. Drag the point marked "a" to explore this problem: what is the full volume of your glass?

3. For a glass with total height H, and stem height h, top radius R and radius of glass base r, can you see that the volume of liquid in the glass requires the calculation of the volume of a frustum - the difference of TWO inverted cones?

Give the formula for the volume of the full glass.

4. Try and find an algebraic model for the volume of drink in the glass as the height varies.

volume(x) =

5. Now use the other drag points to try different glasses!

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Steve Arnold, 2018, Created with GeoGebra

 This problem may also be modelled using CabriJr on the TI-83/84 series calculator. Click on the graphic below to download the file and try it yourself! Try having students work in pairs: one generating data points using the CabriJr model and the partner entering these into lists to plot and then to help build their algebraic model! And just wait until you try this with TI-nspire CAS!

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