Question : Three circles of radius 7 cm are kept touching each other. The string is tightly tied around these three circles. What is the length of the string?
Option 1: $42 + 7\pi\ \text{cm}$
Option 2: $21\pi + 14\ \text{cm}$
Option 3: $42 + 14\pi\ \text{cm}$
Option 4: $7 + 14\pi\ \text{cm}$
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Correct Answer: $42 + 14\pi\ \text{cm}$
Solution :
Given: The radius of the circles($r$) = 7 cm
Length of the string not in contact with circle = $2r+2r+2r= 6r = 42\ \text{cm}$
By symmetry, the angles swiped by a string on one circle = $120°$
Length of string touching the circle = $3\times \frac{120^\circ}{360^\circ}\times 2 \pi r= 2\pi r=14\pi\ \text{cm}$
Total length of string = $42 + 14\pi\ \text{cm}$
Hence, the correct answer is $42 + 14\pi\ \text{cm}$.
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