All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
The hour-glass (Posted on 2009-01-20) Difficulty: 2 of 5
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?

See The Solution Submitted by pcbouhid    
Rating: 2.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 1 of 3

As the top cone has its apex at its bottom and the bottom cone has its apex at its top, the empty portion of the bottom cone is congruent to the still-full portion of the top cone, whose sand will eventually fill in that remaining empty portion on the bottom.

So for the height of the bottom's sand to equal half the height of the sand in the top cone, it must be half the height of its own unfilled portion--that is, 1/3 the height of the cone.

So the sand that has not yet fallen fills a cone similar to the original full cone but 2/3 the size in each linear measure. As a result, it has 8/27 the volume of the original, so 8/27 of the 1.5 hours still remains, and that is 26 minutes, 40 seconds.

Subtracting 0:26:40 from 1:30:00 gives 1:03:20 as the length of time required to reach the state where the bottom sand level is half that in the top--that is, 1 hour, 3 minutes and 20 seconds.


  Posted by Charlie on 2009-01-20 12:43:59
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information