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

Answer 1

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

Answer 2

To find out how many centimeters the needle travels in one minute, we need to calculate the circumference of the record.

The formula for the circumference of a circle is C = π * d, where C is the circumference and d is the diameter.

In this case, the diameter of the record is given as 40 centimeters. So, we can substitute this value into the formula and calculate the circumference:

C = π * 40 = 40π

Now, since the needle makes 33 1/3 circuits around the record in one minute, we can multiply the circumference by the number of circuits to find the total distance traveled by the needle in one minute:

Distance traveled = 33 1/3 * 40π = (100/3) * 40π = 1333 1/3π

So, the needle travels approximately 1333 1/3π centimeters in one minute.

Now, we can look at the answer choices provided:
A. 400
B. 1333 1/3
C. 2000
D. 2400
E. 2666 2/3

Approximating the value of π to be 3.14, we can calculate the approximate value of 1333 1/3π:

1333 1/3 * 3.14 ≈ 4199.67

Since none of the answer choices match the calculated value, we can conclude that there is an error in the given answer choices.