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(A) The fundamental frequency $f$ of a vibrating string is given by the formula $f = \frac{1}{2l} \sqrt{\frac{T}{\mu}}$,where $l$ is the length,$T$ is

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$6: 3: 2$

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(A) The fundamental frequency $f$ of a vibrating string is given by the formula $f = \frac{1}{2l} \sqrt{\frac{T}{\mu}}$,where $l$ is the length,$T$ is the tension,and $\mu$ is the linear mass density.Since the string is the same,the tension $T$ and mass per unit length $\mu$ are constant for all segments.Therefore,the fundamental frequency is inversely proportional to the length of the segment: $f \propto \frac{1}{l}$ or $l \propto \frac{1}{f}$.Given the ratio of fundamental frequencies is $f_1 : f_2 : f_3 = 1 : 2 : 3$.The ratio of the lengths of the segments is $l_1 : l_2 : l_3 = \frac{1}{f_1} : \frac{1}{f_2} : \frac{1}{f_3}$.Substituting the given values,we get $l_1 : l_2 : l_3 = \frac{1}{1} : \frac{1}{2} : \frac{1}{3}$.To simplify this ratio,multiply each term by the least common multiple of the denominators $(6)$: $l_1 : l_2 : l_3 = (1 	imes 6) : (\frac{1}{2} 	imes 6) : (\frac{1}{3} 	imes 6) = 6 : 3 : 2$.

$A$ string is divided into three segments such that the segments possess fundamental frequencies in the ratio $1: 2: 3$. Then,the lengths of the segments are in the ratio:

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