How do I calculate the diameter of a cardboard roll?

Question

A cardboard roll with 25 metres of 0.01mm thick ribbon wrapped around it to give a complete diameter of 10CM

Answer 1

A simpler route to the answer is to take a slice though the roll and consider its area.

The edge of the ribbon has area 25m times 0.01 mm

which is 25,000 mm times 0.01 mm

which is 250 square mm.

The final roll is to have diameter 10 cm, radius 5 cm or 50 mm, therefore sectional area (pi times r-squared) of 2500 times pi square mm.

So the cardboard roll has sectional area of (2500pi -250) square mm, which comes out as 7603.98 square mm.

Divide by pi and take square root, giving a radius of 49.1978 mm which is equivalent to a diameter of 9.83956 cm.

Answer 2

I interpret your questionto to be:
The diameter of a cardboard roll with 25 meters of ribbon 0.01 mm thick wrapped around it gives a complete diameter of 10 cm. You wish to know the diameter of the cardboard roll without the 25 meters of ribbon wrapped around it. First I am assumming that the ribbon is the same width as the cardboard roll, so that the complete diameter is the same everywhere along the tube, and not just where a narrow strip of ribbon is wrapped.

Let d= diameter of cardboard tube in cm.
he circumference of cardboard tube = pi*d cm
So 25 meters (2500 cm) of ribbon will encircle the cardboard tube 2500/(pi*d) times. For each encirclement it will add 0.01mm*2 = 0.02 mm = 0.002 cm to the diameter.
So amount diameter is increased by ribbon is [2500/(pi*d)]*0.002 = 5/(pi*d) cm.
Thus new diameter is 5/(pi*d) +d = 10cm
Simplifying gives pi*d^2-10*pi*d+5=0
Use quadratic formula to give:
d= [10pi +/- sqrt(100pi^2 -4*pi*5)]/2pi
d= (31.41 +/- 30.4)/6.283
d= (31.41+30.4)/6.283= 9.84 cm

Answer 3

Your answer would depend on how long the ccardboard woll is, because this determines how many times the 25 meters of ribbon will wrap anound the roll and then you can tell the thickness of the ribbon and then you ccan find the diameeter of the caardboard roll.

Answer 4

I would save time and go for an approximation:
N = No of turns = 2500/10*Pi
D = Diameter = 10 - 2*N*0.001

This ignores the slight change in length round the roll due to the accumulating thickness of the ribbon.
Pi = 3.14159265358979323846264338327950288419716939937510582097494459
You decide the accuracy to work to.

Answer 5

the circumference of the roll with the ribbon is 0.1Pi (Pi x diameter, in metres) if we unroll the ribbon one rotation, the new diameter is 0.1m-.00002m, and the new circumference is Pi x 0.09998m

We need to know how many times the ribbon is wrapped arround the tube. So we need to add together all the circumferences until we get to 25m.

a formula for the sum of all the circumferences is:
Pi x (0.1+ (0.1-.00002) + (0.1-.00004) +... + (0.1-(n-1)0.00002))
where n is the total number of revolutions.
= Pi x( 0.1n -(.00002+.00004+....+(n-1)0.00002))
=Pi x (0.1n-0.5n(.00002+(n-1)0.00002))
=Pi x (0.1n-.00001n-n(n-1)(.00001))
=-0.00001 Pi n^2+0.1Pi n
Now this must equal 25m, so we can use the quadratic formula to get the results n = 80.2.

So... the ribbon is wrapped round 80 times, contributing 80x0.02mm to the diameter. this means the diameter of the roll is 10-1.6=8.4cm

Answer 6

Calculus for an accurate answer or use piD geometry for an approximate answer where D is the full roll diameter to give circumference which will divide into 25m to give number of layers approx to deduct from original thickness.

Answer 7

i didn't think this could be done without a height but as it turns out it can. i may be wrong but here goes

volume of tube with paper = pi*r^2*H

volume of the paper = H*W*L

therefore volume of tube = pi*r^2*H - H*W*L

the volume of a tube is pi*r^2*h

so to get the radius its

R= sq root((pi*r^2*H-H*W*L)/(pi*H))

the H's cancel out and you are left with

R= sq root((pi*r^2-W*L)/(pi))

then to get the diameter you multiply by 2

W and L are width and length of the paper.

i hope this helps

Answer 8

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Answer 9

pye i think!