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 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 answer:

x =  58.2037 m

Step-by-step explanation:

$$\begin{gathered} D=20\text{cm}\to \text{m}=20:100\text{m}=0.2\text{m} \\ s=30\text{cm}\to \text{m}=30:100\text{m}=0.3\text{m} \\ {s}_0=1\text{cm}\to \text{m}=1:100\text{m}=0.01\text{m} \\ n=80 \\ {x}_0=3.54\text{m} \\ {n}_0=s/s_0=0.3/0.01=30 \\ {r}_0=D/2=0.2/2=\frac{1}{10}=0.1\text{m} \\ {r}_1=r_0+s_0=0.1+0.01=\frac{11}{100}=0.11\text{m} \\ {r}_2=r_1+s_0=0.11+0.01=\frac{3}{25}=0.12\text{m} \\ {x}_1=2\pi \cdot r_0\cdot n_0=2\cdot 3.1416\cdot 0.1\cdot 30\doteq 18.8496\text{m} \\ {x}_2=2\pi \cdot r_1\cdot n_0=2\cdot 3.1416\cdot 0.11\cdot 30\doteq 20.7345\text{m} \\ {x}_3=2\pi \cdot r_2\cdot (n-2\cdot n_0)=2\cdot 3.1416\cdot 0.12\cdot (80-2\cdot 30)\doteq 15.0796\text{m} \\ x=x_0+x_1+x_2+x_3=3.54+18.8496+20.7345+15.0796=58.2037\text{m} \end{gathered}$$

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