|
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)
|
