0

Gramophone tape (plate) with a recorded song with duration $3$ $minutes$ $14$ $seconds$. The playback speed of recording on the gramophone $45$ $rpm$ (revolution per minute). The diameter of the plate is $6.57$ $inches$, and the diameter of the label is $2$ $inches$. The track width is $0.012$ $inches$. How many tracks are there on this record?

My solution is: $6.57$ $inches$ - $2$ $inches$ = $4.57$ inches for tracks. $4.57\over 0.012$ = $380.8333$ is $2$x tracks, then I divide by $2$ and get $191.41666$ tracks on plate and I stuck here. Any help?

Rami Chasygov
  • 248
  • 1
    There's only one track on each side. It wouldn't work, otherwise. –  Dec 15 '17 at 13:19
  • I'm assuming the real question is "how many songs are on this record?" Is that correct? – John Hughes Dec 15 '17 at 13:37
  • @ProfessorVector: Not always - https://en.wikipedia.org/wiki/Multisided_record – Henry Dec 15 '17 at 13:39
  • @JohnHughes, I'm not sure, but probably yes, there means songs – Rami Chasygov Dec 15 '17 at 14:23
  • The assumption that the layers are all cylindrical is a good first approximation. I would use an archimedean spiral.See https://math.stackexchange.com/questions/1633704/the-length-of-toilet-roll or https://www.intmath.com/blog/mathematics/length-of-an-archimedean-spiral-6595 or https://math.stackexchange.com/questions/424669/archimedean-spiral-arc-length-of-coil – cgiovanardi Dec 15 '17 at 20:01

1 Answers1

0

Hint: Suppose you recorded one 3-minute, 14 second song. How many revolutions would that involve? OK, so how many grooves would it involve? [Hint: these numbers are the same.] Now...how many 3-minute songs can you fit into your 191 tracks?

BTW: nice feint on the part of the question-writer including an approximation of $\pi$ (in the form of 3:14, the length of the tune in minutes/seconds format) in the question. :)

John Hughes
  • 93,729
  • 141 resolution takes for 3 minutes 14 seconds song. If one resolution takes one track and I have 191 tracks then there left 41 tracks. Is mean 141 tracks for one song and 41 for another? – Rami Chasygov Dec 15 '17 at 13:56
  • I have no idea what the question-asker meant, but if I were buying the record album, and someone cut off the second song after 1 minute, continuing it on the other side, I'd be annoyed. I think that there's room for one song per side. – John Hughes Dec 15 '17 at 15:14