Tuesday, June 03, 2014

These Are Not the Volumes You're Looking For

A friend, William Ricker, pinged me on a G+ post, which contained the following image, credited to Nathan W. Pyle:

I immediately shared it and replied, "You know that I'll have to calculate the value of d which makes that true." But, of course, I messed up and commented on my own post and not his. No matter. I went ahead and figured it out.

A sphere has a Volume of (4/3) pi * r ^ 3. A hemisphere would therefore be (2/3) * pi * r ^ 3.
Factoring out r ^ 2 from this leaves d ^ 2 = (2/3) * pi * r, meaning that d = SQRT( (2/3) * pi * r), which wasn't as bad as I might have expected.
The problem is that d = 2r, or r = (1/2)d. Which means that I had to start over because this is actually a stinkin' quadratic equation. It was not the expression I was looking for.

Anyway, long story shorter, given the limitations of this text medium, here's the answer: r = 0 (discard) or r = pi / 6, which makes d = pi / 3.

The check is left as an exercise to the reader.


William Ricker said...

Your having derived the droid's diameter as pi/3 leaves the question of what Units this imperial measure uses. Since it's reported the original R2 unit is 18.25 or 18.3 inches American, we can get 3 digit estimate of the size of the imperial unit, 18.3*2.54/(pi/3)=44.3cm.

Why they'd build their droid ~ 5% larger than an imperial unit I don't know. Maybe it's an imperial unit inside diameter?

(x, why?) said...

Despite the fact that they're Rebel droids running from the Empire, they're manufacture actually predates the Empire, so they would be in Republic units. Or possibly Galactic units.

I'd have to consult a different wiki than I usually do for that information.

It's possible that someone in the Galactic Senate decided to pass a law that pi = 3, and then pi/3 would = 1, which would make the R2 unit and actual unit.