Need an algebraic formula to know how much material (in feet) is left in a roll, if all I have is a diameter?

Question

I am working with carpet rolls of different thicknesses. Please Help!!!

I tried im2_weird's suggestion. It does work to some extent, Here's what I came up with, L=(Pi(r^2))/4, in which 4 is the width of the carpet roll (not the diameter). Therefore, if the diameter is 36", r=18", Pi=3.14, then L=254.34 feet. However, according to my measurement, it is actually 276 feet. Too big of a discrepancy, any other thoughts im2_weird?

Thanks all for you help, but specially to psmurty2000, you had the right answer. Just one clarification on his answer, which I think it is just a simple typo. The formula should not be L=pi*d^2/(4*t), but instead it should be L=(pi*r^2)/(4*t), Now the volume formula given by psmurty2000 can also be used to remove the amount taken by the hollow core. Thanks again buddy, you sure know your stuff.

Answer 1

Diameter alone is not enough. You need the to know the thickness also.

Assuming that diameter is 'd' and thickness is 't',

Let r = d/2

Then Volume is V = pi*r^2 *h
Dividing by thickness, Area = V/t
Therefore Length, L = area/h

So, the formula you wanted is L = pi*d^2 / (4*t)

If 'd' and 't' are measured in feet, then 'L' also will be in feet.

Answer 2

Okay, this is my best guess:

I'm going to use an example with a hollow tube in the center b/c it is easier for me to visualize (of course if there is none, the radius of this circle would be 0)

Imagining the cross-section which would be two concentric circles, the distance one 'lap' of carpet travels (the length) would be equal to the circumference of the circle it makes around the tube.

so the 1st 'lap' is simply the circumf. of the tube, the 2nd the cirumf. of the tube plus one thickness of carpet, the 3rd = tube + 2 thicknesses, etc. , so

ℓ = 2πr¹ + 2πr² + ... + 2πr↑(n-1),
where n = # of 'laps' = (radius of carpet roll - radius of tube) / thickness of carpet

ℓ = 2π [ tube radius + (n-1)(carpet thickness)]

But after all this and assuming very accurate measurements, consistency in c.t., no space between layers, etc. maybe would be easier to get an eye for it? lol good luck

Answer 3

Area of a circle (carpet rolled up) is pi * radius^2. Area of a rectangle (carpet laid flat) is length * thickness. The area is the same, so you measure it twice and set equal to one another. Diameter equals twice the radius.

If it helps, you can measure its thickness rolled up.

Hope this helps! If not, perhaps I could have more info?

Answer 4

I just followed his proof through and I agree with psmurty2000. There may be some minor errors that dont account for the hollow air cylinder in the center but this is likely to be very minor. Without getting calculus involved I think this is your best answer.