The coil

How many ropes (the diameter 8 mm) fit on the coil (threads are wrapped close together) The coil has dimension: the inner diameter 400mm, the outside diameter 800mm and the length of the coil is 470mm

Result

x =  2768.529 m

Solution:

$D=8 \ \text{mm} \ \\ D_{1}=400 \ \text{mm} \ \\ D_{2}=800 \ \text{mm} \ \\ l=470 \ \text{mm} \ \\ \ \\ r_{1}=D_{1}/2=400/2=200 \ \text{mm} \ \\ r_{2}=D_{2}/2=800/2=400 \ \text{mm} \ \\ \ \\ S_{1}=\pi \cdot \ r_{1}^2=3.1416 \cdot \ 200^2 \doteq 125663.7061 \ \text{mm}^2 \ \\ S_{2}=\pi \cdot \ r_{2}^2=3.1416 \cdot \ 400^2 \doteq 502654.8246 \ \text{mm}^2 \ \\ \ \\ V=(S_{2}-S_{1}) \cdot \ l=(502654.8246-125663.7061) \cdot \ 470=177185825.662 \ \text{mm}^3 \ \\ \ \\ S_{3}=D^2=8^2=64 \ \text{mm}^2 \ \\ \ \\ x_{1}=V/S_{3}=177185825.662/64 \doteq 2768528.526 \ \text{mm} \ \\ x=x_{1} / 1000=2768528.526 / 1000 \doteq 2768.5285 \doteq 2768.529 \ \text{m}$

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