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Stan's Cassette (Posted on 2004-04-27) Difficulty: 3 of 5
Stan likes music, and likes oldies the best. In his collection of music he has a cassette tape of “The Cassettes 20 Greatest Hits”. The amazing thing about the 20 Greatest Hits, is that each hit is exactly 3 minutes long. This fits perfectly on a 60-minute tape.

Songs 1 – 10 are on Side A, while 11 – 20 are on Side B. When listening to the tape, Stan does one strange thing. He listens to the first 5 songs on side A, then flips it to the other side. He then listens to Side B from that point until the tape ends. How many songs does Stan hear?

  • The tape is wrapped around two spools that have identical diameters of 1 cm.
  • The spool that is receiving the tape rotates at a constant speed.
  • The tape, when located all on one spool, measures 5 cm in diameter.

  Submitted by Leming    
Rating: 3.0909 (11 votes)
Solution: (Hide)
Stan listens to 8 songs. (1,2,3,4,5,18,19,20)

The spool is 0.5 cm in radius. The tape is 2.5 cm in radius around the spool. Total tape cross section on a full spool is pi*(2.5 cm)^2 - pi(0.5 cm)^2. This is 6.25pi cm2 – 0.25pi cm2 = 6pi cm2.

Stan flips the tape after 5 songs, or 50% through Side A of the tape. Since the receiving spool rotates at a constant speed, it should have completed half of its rotations and grown to half the final tape diameter. So it will be the average of the radius of the full spool and that of an empty spool (2.5cm+0.5cm)/2 = 1.5cm.

Cross section of tape on the receiving spool after Hit #5 is pi(1.5cm)^2 – pi(0.5 cm)^2 = 2.25pi cm^2 – 0.25pi cm^2 = 2pi cm^2.

This leaves 6pi cm2 – 2pi cm2 = 4pi cm2 on the original spool.

The radius of the tape on the original spool is the based on the remaining tape cross-section and the spool cross-section. sqrt((4pi cm2 + 0.25pi cm2)/pi) = 2.0615 cm.

Then the ratio of the remaining tape diameter to the original tape diameter gives (2.0615 cm – 0.5) / (2.5 – 0.5) = 78%. Since each song takes 10% or the time, the song heard when the tape is flipped is Hit #18.

Stan listens to 8 songs. (1,2,3,4,5,18,19,20)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Hi im New!leigh berllaque2004-07-28 04:39:04
re: Overcomplicated?Charlie2004-04-29 13:04:21
Overcomplicated?Lisa2004-04-29 11:28:42
solutionKrishneel Nair2004-04-28 02:38:37
re: the real solutionSam2004-04-28 01:59:32
SolutionNo SubjectSteve Royer2004-04-27 22:38:38
Solutionthe real solutionbrad2004-04-27 22:26:09
Solution?Rebecca2004-04-27 22:18:22
re: SolutionCharlie2004-04-27 20:41:24
Charlie does it againLarry2004-04-27 19:52:31
SolutionSolutionTristan2004-04-27 19:33:05
re: solutionBryan2004-04-27 18:28:35
Solution!Lisa2004-04-27 17:24:32
SolutionsolutionCharlie2004-04-27 15:46:49
Some Thoughtsre: assumptionsBUM2004-04-27 15:37:26
SolutionassumptionsLarry2004-04-27 15:15:58
Hints/Tipstoo much info!BUM2004-04-27 15:14:59
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