# 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?

Result

x =  22.39 m

#### Solution:

$D=20 \ cm \rightarrow m=20 / 100 \ m=0.2 \ m \ \\ s=30 \ cm \rightarrow m=30 / 100 \ m=0.3 \ m \ \\ s_{0}=1 \ cm \rightarrow m=1 / 100 \ m=0.01 \ m \ \\ \ \\ n=80 \ \\ x_{0}=3.54 \ \text{m} \ \\ n_{0}=s/s_{0}=0.3/0.01=30 \ \\ \ \\ r_{0}=D/2=0.2/2=\dfrac{ 1 }{ 10 }=0.1 \ \text{m} \ \\ r_{1}=r_{0}+s_{0}=0.1+0.01=\dfrac{ 11 }{ 100 }=0.11 \ \text{m} \ \\ r_{2}=r_{1}+s_{0}=0.11+0.01=\dfrac{ 3 }{ 25 }=0.12 \ \text{m} \ \\ \ \\ x_{1}=2 \pi \cdot \ r_{0} \cdot \ n_{0}=2 \cdot \ 3.1416 \cdot \ 0.1 \cdot \ 30 \doteq 18.8496 \ \text{m} \ \\ x_{2}=2 \pi \cdot \ r_{1} \cdot \ n_{0}=2 \cdot \ 3.1416 \cdot \ 0.11 \cdot \ 30 \doteq 20.7345 \ \text{m} \ \\ x_{3}=2 \pi \cdot \ r_{2} \cdot \ (n-2 \cdot \ n_{0})=2 \cdot \ 3.1416 \cdot \ 0.12 \cdot \ (80-2 \cdot \ 30) \doteq 15.0796 \ \text{m} \ \\ \ \\ x=x_{0}+x_{1}+s_{2}+s_{3}=3.54+18.8496+s_{2}+s_{3} \doteq 22.3896 \doteq 22.39 \ \text{m}$

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