Winch drum

Originally an empty winch drum with a diameter of 20 cm and a width of 30 cm on the rescue car, he started winding a rope with a thickness of 1 cm beautifully from edge to edge. The winch stopped after 80 turns. It remains to spin 3.54m of rope (without hook). How long is the whole rope?

Correct result:

x =  22.3896 m

Solution:

$$\begin{gathered} D=20\text{ cm}\to \text{ m}=20\mathrm{/}100\text{ m}=0.2\text{ m } \\ s=30\text{ cm}\to \text{ m}=30\mathrm{/}100\text{ m}=0.3\text{ m } \\ {s}_0=1\text{ cm}\to \text{ m}=1\mathrm{/}100\text{ m}=0.01\text{ m } \\ n=80 \\ {x}_0=3.54\text{ m } \\ {n}_0=s\mathrm{/}{s}_0=0.3\mathrm{/}0.01=30 \\ {r}_0=D\mathrm{/}2=0.2\mathrm{/}2={\displaystyle \frac{1}{10}}=0.1\text{ m } \\ {r}_1=r_0+s_0=0.1+0.01={\displaystyle \frac{11}{100}}=0.11\text{ m } \\ {r}_2=r_1+s_0=0.11+0.01={\displaystyle \frac{3}{25}}=0.12\text{ m } \\ {x}_1=2\pi \cdot r_0\cdot n_0=2\cdot 3.1416\cdot 0.1\cdot 30\doteq 18.8496\text{ m } \\ {x}_2=2\pi \cdot r_1\cdot n_0=2\cdot 3.1416\cdot 0.11\cdot 30\doteq 20.7345\text{ m } \\ {x}_3=2\pi \cdot r_2\cdot (n-2\cdot n_0)=2\cdot 3.1416\cdot 0.12\cdot (80-2\cdot 30)\doteq 15.0796\text{ m } \\ x=x_0+x_1+s_2+s_3=3.54+18.8496+s_2+s_3=22.3896\text{ m} \end{gathered}$$

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