Half a Glass Please! - GeoGebra Dynamic Worksheet

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. 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 visually accurate model, but it suits for our purposes here!).

Assessment

Be sure you have activated CAS before you begin.

Hint: When entering mathematical expressions in the math boxes below, use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.

Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". More?

1. Drag the slider point marked 'a' to explore this problem. What is the full volume of your glass?

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2. Use the point marked 'a' to find at what depth the glass is half full?

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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 - one an imaginary 'stem cone' of height 10 cm and radius 0.5 cm, and the other with height from the base of the glass to the level of wine in the glass? First, calculate the volume of the 'stem cone' described. stemvol =

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4. If we let 'x' represent the actual level of wine in the glass, then the height of the top of the wine to the base of the glass is given by a function, height(x) =

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5. Using similar triangles, or otherwise, find an expression for the radius of the top of the wine in the glass, in terms of x. radius(x) =

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6. Finally, find an algebraic model for the volume of drink in the glass as the height varies. volume(x) =

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How difficult did you find this activity? Please assign a rating value from 1 (very easy) to 5 (extremely difficult). Comments? Suggestions? What did you learn from this activity?

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