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The pencil (Posted on 2004-02-23) Difficulty: 3 of 5
I have a pencil that always rolls around on a slanted surface. One end is wider and heavier than the other. So whenever it rolls, it goes in a wide circle. Otherwise, the pencil is featureless, only becoming steadily wider towards one end.

The difference between the two diameters on the two ends is exactly 144 times smaller than the length of the pencil. If the pencil is pointing uphill on a slanted surface, how many times will it spin until it points downhill?

  Submitted by Tristan    
Rating: 4.0000 (6 votes)
Solution: (Hide)
The "length" of the pencil is actually the length from the middle of the two ends of the pencil. The length that touches the table is actually slightly larger. The radii difference is 288 times smaller then the "length," but the length we want is √(1+288^2) times larger. This is about 288.0017361. The difference in diameters is 144.0008681 times smaller than the length that touches the table.

When the pencil rolls, the two ends make two semi-circle paths. The length of the two paths are proportional to the circumferences of the ends of the pencil. The proportion is equal to the number of spins. The difference between the radii that make the circles is equal to the length of the pencil on the table.

Let s=the number of spins, d=the diameter of the smaller end, r=radius of the smaller semi-circle path, and L=length touching table. So put together, s*d*π=r*π for both larger and smaller diameters and radii. The difference in diameters is 144.0008681 times smaller than L. Solving, we get s*d = r and s(d+L/144.0008681)=(r+L).
r/d=(r+L)/(d+L/144.0008681)
r*(d+L/144.0008681)=d*(r+L)
r*L/144.0008681=d*L
r/d=144.0008681
r/d=s=144.008681

Comments: ( You must be logged in to post comments.)
  Subject Author Date
simply put solutionDan Porter2004-02-28 22:39:58
re(2): But on the other hand....Penny2004-02-28 08:59:49
re: But on the other hand....nikki2004-02-26 17:47:21
re: But on the other hand....Brian Wainscott2004-02-24 14:57:06
But on the other hand....Penny2004-02-24 14:18:03
re(2): Trying to be even more exactThalamus2004-02-24 12:57:25
Hints/Tipsre(6): engl. unutsAdy TZIDON2004-02-24 11:53:59
re(5): Trying to be even more exact - dumb questionfwaff2004-02-24 11:02:30
re(4): Trying to be even more exactCharlie2004-02-24 10:53:25
re(3): Trying to be even more exactfwaff2004-02-24 10:49:17
re(2): Trying to be even more exactPenny2004-02-24 09:44:52
re: Trying to be even more exactCharlie2004-02-24 08:21:02
My thoughtRoberto Mattos2004-02-23 17:59:53
SolutionTrying to be even more exactPenny2004-02-23 17:15:04
SolutionTrying to be exact.Charlie2004-02-23 15:29:48
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