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

Correct result:

x =  2768.5285 m

Solution:

$$\begin{gathered} D=8\text{ mm } \\ {D}_1=400\text{ mm } \\ {D}_2=800\text{ mm } \\ l=470\text{ mm } \\ {r}_1=D_1\mathrm{/}2=400\mathrm{/}2=200\text{ mm } \\ {r}_2=D_2\mathrm{/}2=800\mathrm{/}2=400\text{ mm } \\ {S}_1=\pi \cdot r_1^2=3.1416\cdot 200^2\doteq 125663.7061{\text{mm}}^2 \\ {S}_2=\pi \cdot r_2^2=3.1416\cdot 400^2\doteq 502654.8246{\text{mm}}^2 \\ V=(S_2-S_1)\cdot l=(502654.8246-125663.7061)\cdot 470=177185825.662{\text{mm}}^3 \\ {S}_3=D^2=8^2=64{\text{mm}}^2 \\ {x}_1=V\mathrm{/}{S}_3=177185825.662\mathrm{/}64\doteq 2768528.526\text{ mm } \\ x=x_1\mathrm{/}1000=2768528.526\mathrm{/}1000=2768.5285\text{ m} \end{gathered}$$

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