Distance

[Guest]

The first turn of the record will have a 5 inch radius....find the circumference
The next turn will have a radius 1/90 inch less find its circumference and add it on....
keep on doing this for each turn and add them all together.......
for 3 inch travel radially there will be 270 turns and 270 calculations needed....and add them all up and see if comes to 495 feet.

That ia a lot of calculations.....so this is why I dis it by the area method earlier......OR it can be done using the sum of N terms in a series

[Guest]

"OR it can be done using the sum of N terms in a series " I have no idea.

[Guest]

are you not going to try any method of calculating it?

[Guest]

For the series calculation you will have to figure out is there a common difference between each groove around the record...is so it will form an arithmetic series.
OR if the length adjacent grooves have a common ratio......then the will form a geometric series

Look up Arithmetic series or Geometric series on internet...how to do them. sometimes they are called arithmetic and geometric progressions

[Guest]

I don't know which to use. I'll just use your solution for area. Thanks again.

[Guest]

This is both solutions together.... one based on Area of recording and other based on Sum of N terms in a series of circles.

The original question.............

A phonograph record is 12" across.
An outer non-playing area 1" in width.
A non-playing central area 4" in diameter.
An average of 90 grooves per in.

Determine distance the needle travels when the record plays once.

Assume this means the distance travelled along the record track or groove.
......................................

Solution based on area of recording and considering the groove as a long strip

Consider the groove as a long strip (1/90) inches wide.
The disk is 5" outer and 2" inner radius

Area of disk is pi x (5^2 - 2^2)
= pi x (25 - 4)
= pi x 21

length of groove = pi x 21 / (1/90) / 12 ..... expressed in feet
= pi x 21 x 90 / 12
= 22/7 x 21 x 90 / 12
= 66 x 90 / 12
= 11 x 90 / 2
= 11 x 45 = 495 feet

......................................
........................................

Solution based on arithmetic series.............

Now consider the recording as a spiral groove ... round and round the record.............

As the record rotates the groove moves from a radius of 5 inches to a radius of 2 inches.
That is a radial distance of 3 inches and the grooves are spaced at 90 groover per inch
so that means 3 x 90 = 270 grooves OR 270 circles OR 270 rotations to play the recording once.

We have to find the total length of all of these circles to find the length of the recording

The diameters of the outside circle will be (12 - 2) = 10 inches (as stated in the question) for the first rotation, then the diameter of the next rotation will be 10 - (2/90) inches ( ie. the width of 2 grooves less), and the diameter of the 3rd rotation will be 10 - (4/90) inches.......and so on until we get to the 270 rotations when the diameter will be 10 - (540/90) inches OR 10 - 6 inches OR 4 inches as stated in the question.
So we have an arithmetic series, first term is 10, common difference is minus(2/90) and there are 270 terms
So we need to find the sum of the first 270 terms to find the total lengths of the diameters and multiply this by Pi to get the total circumferences.

The formula for the sum is as follows......... you can look it up on the internet etc.....

S = (n/2)(2a - d(n-1))

S = (270/2)(2x10 - 2/90(269))

= 135(20 - 5.978)
= 135 x 14.022 = 1892.97 ....sum of the diameters
now multiply by Pi = 22/7 to get circumferences
= 22/7 x 1892.97 = 5949.33
= 5949.33 / 12 ....to convert to feet
= 495.8 feet
...........................

Both answers correspond ............
...............................................

[Guest]

to clarify the formula.......

S = (n/2)(2a + d(n-1)) ...this is the standard formula ....


it should have be written as .... S = (n/2)(2a + d(n-1))

and in our case as "d" is negative it becomes......S = (n/2)(2a - d(n-1))

and putting in the values gives.......... S = (270/2)(2x10 - 2/90(269))

[Guest]

"length of groove = pi x 21 / (1/90) / 12 ..... expressed in feet
= pi x 21 x 90 / 12
= 22/7 x 21 x 90 / 12
= 66 x 90 / 12
= 11 x 90 / 2 **
= 11 x 45 = 495 feet"

** I am not clear on this step.


I am not totally clear on the series steps but I understand more than I did.

[Guest]

= 66 x 90 / 12 ........this is 66 multiplied by 90 then divided by 12 ....6 divides into 66 on the top .... 11 times and 6 divides into 12 on the bottom ... 2 times

= 11 x 90 / 2 ** ............ to leave 11 multiplied by 90 on the top then divided by 2 on the bottom.......this is called cancelling or simplifying to lowest terms

= 11 x 45 = 495 feet"

** I am not clear on this step ........you would need to be familiar with adding, subtracting, multipling and division on ordinary number expression before starting to do problems of this complexity. ..... look up questions on simplifying expressions and cancelling fractional expressions to lowest terms etc

[Guest]

Ok - thanks for the additional info and suggestions.