Question : Three circles of radius 6 cm are kept touching each other. The string is tightly tied around these three circles. What is the length of the string?
Option 1: $36 + 12\pi\ \text{cm}$
Option 2: $36 + 18\pi\ \text{cm}$
Option 3: $24 + 36\pi\ \text{cm}$
Option 4: $36 + 20\pi\ \text{cm}$
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Correct Answer: $36 + 12\pi\ \text{cm}$
Solution :
The length of the string not in contact with the circle = $2r+2r+2r=6r=6\times6=36$
By symmetry, the angle swiped by the string on one circle = $120^\circ$
Length of the string touching circle = $\frac{2\pi r}{3}+\frac{2\pi r}{3}+\frac{2\pi r}{3}=\frac{6\pi r}{3} = 2\pi r$
$=2\pi \times 6=12\pi$
$\therefore$ Total length = $36+12\pi\ \text{cm}$
Hence, the correct answer is $36+12\pi\ \text{cm}$.
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