real life math

Friend and I stop for a beer in the afternoon, drawn in by the following deal on a chalkboard:

"Cana Two For One".

It turns out that a cana is priced at $2.50 and a regular old pint is $5.00. The special deal puts the Cana at a quarter of the price of a pint, provided you drink at least two. In fact, you can get no less than four canas for the price of one pint. Excellent? Certainly seems that way, until you consider the size of the cana versus the pint. It gets a little dicier. The cana is freaking tiny. Exhibit A:

(That would be pint on the left, cana on the right.)

So what is the better deal? Four canas or one pint?

Comparing volumes is usually an estimation killer, since, in my experience, the brain is usually pretty bad at estimating volume. This is probably because quadratic relationships are somewhat "unnatural", at least when compared with linear ones.

But we push through. We call each glass a cylinder (imperfect, but it'll have to do). The pint's diameter registers at one of my pointer fingers, almost exactly. Conveniently, my finger is divided neatly (intelligently designed?) into three pretty equal sections; the cana's diameter is almost exactly two-thirds of the same finger. So we can call the radius of the cana two-thirds of the radius of the pint. Similarly, the height of the cana works out to be about two-thirds the height of the pint.

We're good to go. Let's call the pint volume V, the pint radius R, and the pint height H. Similarly, the cana volume will be v, the cana radius r, and the cana height h.

The pint volume is given by:
The cana volume is given by:
We've established relationships between R and r and H and h:

Now we can substitute into the equation for v:

Simplifying the squared business:
Finally multiplying all fractions:
Which leads to final comparison of v and V:
This works out nicely; eight-twenty sevenths is pretty darn close to one-third (nine-twenty sevenths). So we are getting a deal (well, friend is, since I opted out), since four canas equate with the price of one pint, but three canas has already equated the volume in one pint.

Not bad for a Saturday afternoon. Facility with fractions and formulas. We hit quite a few Integrated Algebra performance indicators.