Math CentralQuandaries & Queries


Question from Philiip:

What diameter reel would be required to roll up 1000 meters of tape that is 0.01 mm thick such that the final role left 10 mm empty space between the top of the roll and the outside of the reel, the hub that the tape was wound on was 2 cm?

Hi Philip. Use cross-sectional area to figure this out.

1000 meters length times 0.01 mm thickness gives an area of 10 000 square mm.

Assuming you are telling me the diameter of the hub is 2cm (if that is the radius or the circumference, you will need to make the appropriate adjustment), then the area of the hub is 314 square mm. That's pi times the square of the radius.

That makes a total of 10 314 square mm.

A circle with this area would have a radius of sqrt(10314 / 3.14) = 57.3 mm. And that means we add 10mm to this radius to find the radius of the reel: 67.3mm.

The diameter of the reel is therefore twice this radius, making 134.6 mm.

Stephen La Rocque.

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