I need a maths whiz!

[smartcooky]

I have what seems to be a simple problem but I just cannot see a simple solution.

I have rolls of photographic paper which I use at work. They look similar to this



What I want to do is to be able to measure the diameter of a part roll to estimate the amount of paper left on the roll, given that I know the following...

d_c - the diameter of the core
d_f - the diameter of a full roll
L_f - the number of metres of paper on a full roll

It should be possible to come up with a formula, which I imagine will involve pi somewhere along the line, to do this...

 

[Bryanpoprobson]

If you could gauge the thickness of the paper the formula is..

Length of Paper in Feet:            1000*3.1426*(Do^2 - Di^2) = 65.47*(Do^2 - Di^2)
                                                _______________________     _________________
                                                       (t*12*4)                                        t

Where......
Outer Roll Diameter                            Do
Outer Diameter of Core                      Di
Thickness of Paper                                t

Length of Paper in Meter:            1000*3.1426*(Do^2 - Di^2) = 78.56*(Do^2 - Di^2)
                                                        (t*10*4)                                        t

[Bryanpoprobson]

Quick search comes up with this...

http://www.cutsmart.com/pages/length_on_roll_calculator.html

[Zakalwe]

The tightness of the roll might throw this off a fair bit? Would you be better off doing this by the weight of the roll?

[smartcooky]

If you could gauge the thickness of the paper the formula is..

Length of Paper in Feet:            1000*3.1426*(Do^2 - Di^2) = 65.47*(Do^2 - Di^2)
                                                _______________________     _________________
                                                       (t*12*4)                                        t

Where......
Outer Roll Diameter                            Do
Outer Diameter of Core                      Di
Thickness of Paper                                t

Length of Paper in Meter:            1000*3.1426*(Do^2 - Di^2) = 78.56*(Do^2 - Di^2)
                                                        (t*10*4)                                        t

Right, so it no wonder I cannot work out how to do it using the known length of a full roll and without accurately knowing the thickness paper. However...

Quick search comes up with this...

http://www.cutsmart.com/pages/length_on_roll_calculator.html

This is great because I know the length of paper on a full roll (100m) so I can input the known values of Do and Di and substitute various values for t until I get 100m for an answer. Turns out the paper is .1mm thick

The tightness of the roll might throw this off a fair bit? Would you be better off doing this by the weight of the roll?

Yep, I had thought of that. I carry five different paper widths, 4", 5" 6" 10" and 12", so I would need to record the weight of all five. I'd need to buy scales to weigh the paper.

It does have the advantage of being a "linear" solution; full roll weight minus core weight = 100m, so a roll that weight (-core) of only 35% of full will have 35 metres left.


Thanks for the help guys. I think I'll use the website Bryan found (desktop shortcut). 

[ka9q]

Yeah, I'd do it by weight too. If it's too difficult to weigh all five, and if you can assume they all have the same weight per unit area, then you can just scale by the widths.

[Bob B.]

L = L_f * (d - d_c)² / (d_f - d_c)²

where L if the length remaining and d is the diameter of the partially used roll.

[Bob B.]

Sorry, made a mistake.  Ignore my previous post.  The corrected formula is,

L = L_f * (d² - d_c²) / (d_f² - d_c²)

[smartcooky]

Sorry, made a mistake.  Ignore my previous post.  The corrected formula is,

L = L_f * (d² - d_c²) / (d_f² - d_c²)

Thank you Bob, that works perfectly

Checked against the website Bryan posted, and against a test I did when I used your formula to calculate the length (and therefore the number of 7"x5" prints left) on a roll of 5" paper. Your formula gave 15.6m (therefore 86.6 prints), and it ran out on the 88th print; accurate enough for my purposes.

I'm going to have a crack at programming that into my old HP-25C (if I can remember how!).

[Bob B.]

Thank you Bob, that works perfectly

I'm glad I could help.

[Allan F]

It might be an idea to draw a diagram, and print it out, that way you don't need to do the math every time, and can just look it up on the wall.

[smartcooky]

It might be an idea to draw a diagram, and print it out, that way you don't need to do the math every time, and can just look it up on the wall.

Yes. Another way will be to plug bob's formula into an excel lookup table.