A record playa with a speed of 33 1/3 revolutions per minute, meaning that a needle in a groove around te edge of the record will make 33 and 1/3 circuits around the record in one minute. If the diameter of the record is 40 centimeters, how many centimeters does the needle travel in one minute?

Question

A record playa with a speed of 33 1/3 revolutions per minute, meaning that a needle in a groove around te edge of the record will make 33 and 1/3 circuits around the record in one minute. If the diameter of the record is 40 centimeters, how many centimeters does the needle travel in one minute?
A. 400
B. 1333 1/3
C. 2000
D. 2400
E. 2666 2/3

Reiny · Answer

This is a very strange question.
When I used to have a record player, the needle did not travel around, the record did, but I know what you mean to say.
in one rotation the 40π cm of the groove will travel past the neddle or appr 125.66 cm
so in 1 minute the distance would be 100/3 * 125.66
= appr 4189 cm

The problem with my calculation is that it assumes that the needle maintains the distance of 20 cm from the centre.
But it does not do that, the radius keeps shrinking very slowly.
None of the given answers come even close to my estimation.

This is very complicated question, since you are finding the length of a spiral.

Here is a page showing the real complexity of this problem.
http://www.intmath.com/blog/mathematics/length-of-an-archimedean-spiral-6595