## Tuesday, March 8, 2011

### Coke v. Sprite?

Inspired by Dan Meyer's super-cool WCYDWT problem Coke vs. Sprite, students in physics and honors physics set out yesterday to determine if the Coke glass or the Sprite glass had more of its original drink at the end.  I have the results here and a few observations and extensions (intended for my students, but hey, who's counting?).

In the honors class, there were four predictions of Sprite, three of Coke, and three that they'd be the same.  As they pursued the problem, some opinions changed when confronted with the icy-cold refreshment of the math.

In the physics class, there were shifting predictions for a few minutes after viewing the video, but by the time we split into groups, everyone thought that Sprite was the winner. I put the big emphasis in this section on showing, explicitly, absolutely every step of the logic that got them from the beginning to the end.  I didn't have to tell them if they were right or wrong, though I did point out and wishy-washiness in their logic.  After all, only one conclusion can be logically argued from the video!

Some students (n-1, actually), picked a dropper size and manually calculated the sameness of the final original soda volumes:

Solving a specific instance of a problem is a great first step toward building a general solution, but it really isn't the end.

If I prove that the Titanic can float (well...), that doesn't prove that everything metal or everything red and black or everything with one fake chimney (really!) will float.  A general argument is... general.  An example may be suggestive, but is just an example.  The plural of 'example' is not 'proof.'

A generalized algebraic argument may be made, but here's a super-elegant diagrammatic argument, made by an enterprising young lady in the physics class:
Another group had a similar idea, but used a dropper for the shape:

How about that? A diagram that helps solve the problem.  It seems to me that I might've said something before about diagrams being useful...

Here's another model of a related question that a group in the second honors class came up with.  It shows the amount of Coke/Sprite left in their original glasses as a function of dropper size.  The endpoints weren't too hard to find, though it was yet another clue for modeling:

Always consider the special cases!

What would the final amount be if the drop had a size of zero?  ...of 12 ounces?  Those easy questions reveal the beginning, the end, and something about the shape.  It doesn't tell you everything, though: you need to know if the function's linear or... not.

It turns out that it isn't (it also turns out that I didn't get this picture taken until they'd erased their board and started to work on the next thing, so I recreated it - sorry!).  The group that made this graph used a calculator to find the original-glass percentage for 10 or 12 different dropper sizes (they did it all in a table, making it pretty easy).

Why isn't it linear?  Well, we can go all crazy with the algebra!  If we call the original volume V and the dropper volume n, then, after the first dropper is transferred:

Glass 1 - Sprite: V - n     Coke: 0
Glass 2 - Sprite: n     Coke: V - n

The second dropper's the trick.  The proportion of Sprite that's in the second glass is: (if you don't see the equation in your reader, click the post title to go to the original post - sorry!)
${\color{white} \frac{n}{V+n}}$
The proportion of Coke in the second glass is:
${\color{white} \frac{V}{V+n}}$
The amounts transferred are therefore:
Sprite: ${\color{white} n\left (\frac{n}{V+n}} \right )$     Coke: ${\color{white} n\left (\frac{V}{V+n}} \right )$
This leaves:

Glass 1 - Sprite:   ${\color{white} V-n + n\left (\frac{n}{V+n}} \right )$   Coke:  ${\color{white} n\left (\frac{V}{V+n}} \right )$
Glass 2 - Sprite: ${\color{white} n - n\left (\frac{n}{V+n}} \right )$     Coke: ${\color{white} V - n\left (\frac{V}{V+n}} \right )$

These don't look equal at first glance, but they both reduce (after finding a common denominator) to:
${\color{white} \frac{V^2}{V+n}}$
Check the units:
• Volume-squared over volume: units of volume - check!
Check the special cases:
• If the whole glass is the transfer size, then the glasses should be half-and-half at the end, which the function predicts - check!
• If nothing is transferred, then the glasses should have all of their original soda - check!
The function correctly predicts the special cases, has good units, and all of the expected behaviors.  Now you know that you have a model worth working with!