An hour-glass is formed by two identical cones.
Initially, the top cone is full of sand, and the bottom cone is empty.
The sand starts flowing down at a constant rate and the top cone is emptied in exactly 1:30 hours.
How long does it take for the height of the sand in the bottom cone be half of the height in the top cone?
A very nice and succinct explanation Charlie! Not sure why the "8/27" bit should be so obvious at the outset though (2^3/3^3 perhaps...my geometry's a bit rusty as to why that ratio can be determined up front!). I tried the problem 'longhand' with pencil and graph paper using various radii, cone heights divisible by 3 (to keep the 2/3:1/3 height relationship), calculating the slant angles and volumes, etc. Every one worked out, of course, to 1 hour, 3 minutes and 20 seconds as per your solution, with the volumes of sand in the bottom cone in the end indeed being 8/27 of the full cone volumes in every case.
But I do have one observation/question. The problem basically involves a sinking 'cone' of sand in the top cone, leaving a 'void' (cone fructum) above that essentially should be of equal volume to the fructum of sand growing in the bottom cone. The math in your solution obviously works, but inherently assumes that the sand surfaces (cone or fructum) remain totally flat and parallel to the hourglass bases throughout.
In my book that would really only happen using water instead of sand. Falling sand would typically result in a conic depression in the centre of the top cone, which would then form a conic shape (not totally flat) in the bottom as well. The following crude ASCII illustrates. Water might indeed be represented by the bottom two flat lines of === in the bottom cone, but sand would still form a middle peak suggested by the upper two lines, which compare to the central conic void in the top cone. (The equilibrium slope of the piles likely being some function of the coarseness of the sand itself.)
::::::::::::::::::
\ /
\ /
\== ==/
\== ==/
\====/
\===/
\==/
/. \
/ .. \
/ . \
/ = \
/ === \
/======\
/=======\
::::::::::::::::::
So we might actually be dealing with the height of sand in the middle of the bottom cone in relation to the outer sand height in the top cone. Hm-mm-m! I'm thinking the real solution to this problem might actually involve some physics as well, and be a lot more complicated. (Then again, this would have been a Science problem I guess, or the basic math still applies regardless.)
Edited on January 20, 2009, 5:37 pm
Sand vs. Water?
