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| Math Central | Quandaries & Queries |
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Question from james: I am after a standard calculation so that after each usage of the Reel I can get the Remaining Quantity in kilos left on the roll below is an example of roll THICKNESS: 30 MICRON The core centre has a 9cm diameter, the weight of the roll is excluding the core , and after each use the diameter of the role is measured. I am a factory supervisor working on project and need this calculation info for template.. Thanks James |
James,
The volume of a cylinder is given by π r2 h cubic units where r is the radius, h is the height and π = 3.1416. Suppose R is the radius of the roll, r is the radius of the core and h is the height of the roll. For your example R = 76/2 = 38, r = 9/2 = 4.5 and h = 7.5, all in centimeters. The volume of film on the roll is then
original volume = π R2h - π r2 h = π h (R2 - r2) cubic units.
Suppose the mass is W units (for your example W = 7.13 kilos). The density of the film is then
W/(original volume) mass units/(cubic length units).
For your example this is
7.13/[π × 7.5 (382 - 4.52)] kilos per cubic centimeter.
Suppose that, after a use, the radius of the film remaining on the roll is s units. The volume of film remaining on the roll is then
new volume = π s2h - π r2 h = π h (s2 - r2) cubic units.
Since the density is [W/(original volume)] mass units/(cubic length units) the mass of the reduced roll will be
[W/(original volume)] × (new volume) =
mass units.
For your example that would be 
kilos.
Harley
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