The coil

How many ropes (a diameter of 8 mm) fit on the coil (threads are wrapped close together)? The coil has the following dimensions:
The inner diameter is 400mm.
The outside diameter is 800mm.
The length of the coil is 470mm.

Final Answer:

x =  2768.5285 m

Step-by-step explanation:

$$\begin{gathered} D=8\text{mm} \\ {D}_1=400\text{mm} \\ {D}_2=800\text{mm} \\ l=470\text{mm} \\ {r}_1=D_1/2=400/2=200\text{mm} \\ {r}_2=D_2/2=800/2=400\text{mm} \\ {S}_1=\pi \cdot r_1^2=3.1416\cdot 200^2\doteq 125663.7061{\text{mm}}^2 \\ {S}_2=\pi \cdot r_2^2=3.1416\cdot 400^2\doteq 502654.8246{\text{mm}}^2 \\ V=(S_2-S_1)\cdot l=(502654.8246-125663.7061)\cdot 470\doteq 177185825.6625{\text{mm}}^3 \\ {S}_3=D^2=8^2=64{\text{mm}}^2 \\ {x}_1=V/S_3=177185825.6625/64\doteq 2768528.526\text{mm} \\ x=x_1/1000=2768528.526/1000=2768.5285\text{m} \end{gathered}$$

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