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A 33 1/3 rpm phonograph record of the "1812 Overture" plays 15 minutes and 12 seconds. The music starts at a point 14.5 cm from the center of the record and ends 6.1 cm from the center. How long, to the nearest centimeter, is the groove?
time: 15:12 = 15.2 minutes
number of revolutions 15.2(100/3) = 506.67
change in radius is: 14.5 - 6.1 = 8.4
Change in radius for one revolution: 8.4/506.67 = 0.01658 cm
Average radius, for the first revolution: 14.5 - 0.01658/2 = 14.5 - 0.00829 cm
Average radius, for the second revolution: 14.5 - 0.01658 - 0.00829 cm
Average radius, for the nth revolution: 14.5 - 0.01658(n - 1) - 0.00829 cm
We need to sum these, from n = 1 to n = 506.67
506.67(14.5 + 0.00829) - 0.01658*SUM(n)
= 7350.9 - 2132.4 = 5218.5 cm
Finally multiply by 2pi to get the length of the groove.
Answer: 32789 cm
A 33 1/3 rpm phonograph record of the "1812 Overture" plays 15 minutes and 12 seconds. The music starts at a point 14.5 cm from the center of the record and ends 6.1 cm from the center. How long, to the nearest centimeter, is the groove?
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