Design a Better Drink Can - GeoGebra Dynamic Worksheet

©2019 Compass Learning Technologies ← Live Mathematics on the Web ← GeoGebra Assessment Showcase →Design a Better Drink Can
Design a Better Drink Can
Adapted from an algebraic modelling activity on the former Charles Sturt University HSC Online website.

A standard can of drink holds 375 ml.
Most are around 6cm across the base, and 13 cm high.
But is this the BEST size?
Since the cost of materials is most closely related to the surface area of the can, we would like dimensions that give the greatest volume for the smallest surface area.
Drag the point A shown below to change the radius and observe the graph to see how the surface area (divided by 100!) changes! Try different volumes.
Our task here: To try to find the algebraic function which matches this situation.
Discussion
What assumptions have we made in building this model, and how might these be minimised?
Would a volume of, say, 400 mls make more sense - allowing for some air in the top of the can?
The top and bottom of the can are thicker than the sides - perhaps our surface area formula shold be adjusted to allow for, say, double thickness at top and bottom?

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Assessment

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?

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1. Drag the radius point A to find the radius which gives the smallest surface area for a volume of 375 ml.

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2. Give a formula for the surface area of a can with radius 'radius' cm and height 'height' cm. Surface Area =

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3. To build an algebraic model, we begin with a function for the height of the can, given a volume (v) and radius, x. height(x) =

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4. The function for the surface area, then, for a can of volume v and radius x is given by surfacearea(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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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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Extension...

I decided to extend this problem a little further, and found that I had five different can sizes in my pantry! (Note that my data opposite includes a can of soft drink which is not shown).

Can you pick the odd one out?

Are any of these cans optimal?

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©2019 Compass Learning Technologies ← Live Mathematics on the Web ← GeoGebra Showcase