Question

how long is the groove on one side of a long play(33 1/3 rpm) phonograph record? assume there is a single recording and the author (beginning) groove is 5.75 inches from the center and the inner(ending) groove is 1.75 inches from the center. the recording plays for 23 minutes.

Answer 1

23 min * 100/3 rev/min = 766.666 rev

So, divide the distance between the 1st and last grooves into 767 grooves, and you get

4/766.666 = 0.0052 inches between grooves

So, you have an arithmetic sequence with

a = 5.75*2π
d = 0.0052*2π

and you want S₇₆₇

Just plug and chug

Answer 2

You would have to use 1.75*2pi for a or -0.0052 for d