More on a roll of paper

Math Central

Quandaries & Queries

Can you help with the equations to calculate the length of paper required to
achieve a target outside diameter when wrapped around a core please?

The inside diameter of the cardboard core is 76mm
The thickness of the wall of the cardboard core is 5mm
The thickness of the paper is 138microns
The desired outside diameter of the finished roll is 320mm

I hope you can help me with a solution, Thank you

Hi Stephen,

The inside diameter of the cardboard core is 76 mm so the inside radius of the cardboard core is $\large \frac{76}{2} \normalsize = 38$ mm. If I understand what you mean by "the thickness of the wall of the cardboard core" then the outside radius of the cardboard core is 38 + 5 = 43 mm.

Now look at our response to a previous question from Tuomas. Using the notation in that response you have $r = 43 \mbox{ mm }, R = \large \frac{320}{2} \normalsize = 160 \mbox{ mm },$ and since there are 1000 microns in a millimeter, $t = 0.138$ mm. Now you can use the expression

\[t \times L = \pi \; \left( R^2 - r^2\right)\]

to find $L.$

Write back if you need more assistance.

Harley

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.