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January 11th, 2018, 05:43 PM   #1
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Unique Circle Area problem

I have a roll (circle) that contains widgets (pins/lugs). I know dimensions about this roll of widgets and I'd like to estimate when I have 5000, 3000, 2000, and 1000 widgets left on the roll. Here is how I set up the problem to solve:
If I know how many widgets I have to start (W) and I know the outside radius dimension of the roll (O), and the inside radius dimension where there aren't widgets (I), then I can figure out roughly how many widgets are left based on the area % and solve for (O) the radius.

If I know that the area of the circle when full is pi * r^2 or pi *81=254.5
I can calculate the radius if I know the area by SQRT(area/pi) or SQRT(81) or 9. Which I know is a full roll.
If I have 2000 widgets, then I know that the roll area would be 66% full. SQRT((area*.66)/pi)= r or 7.31 inches from the center.

But.......this is only accurate if the center is filled with widgets. I know that the core is 7.25 in Diameter in this example.
I know that I have to subtract the core, but I'm having a hard time wrapping my head around what that does to the radius number at 3000, 2000, and 1000. I've also included 1 widget as a "gut check" because if my formulas are right, 3000 should be at 9 inches and 1 should be just above 3.625 inches.

How do I solve for the radius of this roll when 66% is used from the OD and the ID stays the same?

I've tried calculating as if the roll was filled to the center (3000 = 3487) and then subtract it out later, but I can't get anything to work.
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January 11th, 2018, 09:28 PM   #2
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How did you calculate 3487?

I get 3580.9 * 0.16223 = 580.9 approximately (because 3000/(1 - 0.16223) = 3580.9 approximately).
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January 12th, 2018, 04:00 AM   #3
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Is that right? Math error on my part maybe. I have 3000 widgets in the image for a full circle and I find that the center core takes up about 16%. 3000 x .016 is 487 imaginary widgets living in the core.

Now look what we've done, we've got imaginary widgets!

But how do I wrap my head around how to calculate a partial radius? When I remove the inner core numbers, things don't add up.

Any help is appreciated.
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