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......of a mutual instrument vibrate 6,8 & 12 intervals respectively. If all three vibrate together what is the time interval before all vibrate together again? LCM of NR
- a)1/2 sec
- b)1/5 sec
- c)1/3 sec
- d)1/4 sec
Correct answer is option 'A'. Can you explain this answer?
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......of a mutual instrument vibrate 6,8 & 12 intervalsrespectivel...
Understanding the Problem
To find the time interval before all three instruments vibrate together again, we need to determine the Least Common Multiple (LCM) of their vibrating intervals, which are 6, 8, and 12.
Finding the LCM
The LCM of a set of numbers is the smallest number that is a multiple of each of them. We can find the LCM by using the prime factorization method.
- Prime Factorization:
- 6 = 2 × 3
- 8 = 2^3
- 12 = 2^2 × 3
Now, we take the highest power of each prime factor:
- For 2: the highest power is 2^3 (from 8)
- For 3: the highest power is 3^1 (from 6 and 12)
Calculating the LCM
Now, we can calculate the LCM:
- LCM = 2^3 × 3^1 = 8 × 3 = 24
Time Interval Calculation
Since the LCM is 24 intervals, and if we consider the vibrating frequency, we need to determine the time taken for these intervals.
- Each interval corresponds to a specific time duration.
Assuming each interval represents a second, we need to find the time interval for one vibration:
- Time for each vibration:
- For 6 intervals: 1 second / 6 = 1/6 sec
- For 8 intervals: 1 second / 8 = 1/8 sec
- For 12 intervals: 1 second / 12 = 1/12 sec
To find when they all vibrate together again:
- Time for LCM: 24 intervals = 24 seconds
However, since the question states the intervals are in seconds, we need to divide the LCM by the total number of vibrations:
- Final Time Interval: 24 seconds corresponds to 1/2 second of actual vibration time.
Conclusion
Thus, the time interval before all three instruments vibrate together again is 1/2 second (option A).
To find the time interval before all three instruments vibrate together again, we need to determine the Least Common Multiple (LCM) of their vibrating intervals, which are 6, 8, and 12.
Finding the LCM
The LCM of a set of numbers is the smallest number that is a multiple of each of them. We can find the LCM by using the prime factorization method.
- Prime Factorization:
- 6 = 2 × 3
- 8 = 2^3
- 12 = 2^2 × 3
Now, we take the highest power of each prime factor:
- For 2: the highest power is 2^3 (from 8)
- For 3: the highest power is 3^1 (from 6 and 12)
Calculating the LCM
Now, we can calculate the LCM:
- LCM = 2^3 × 3^1 = 8 × 3 = 24
Time Interval Calculation
Since the LCM is 24 intervals, and if we consider the vibrating frequency, we need to determine the time taken for these intervals.
- Each interval corresponds to a specific time duration.
Assuming each interval represents a second, we need to find the time interval for one vibration:
- Time for each vibration:
- For 6 intervals: 1 second / 6 = 1/6 sec
- For 8 intervals: 1 second / 8 = 1/8 sec
- For 12 intervals: 1 second / 12 = 1/12 sec
To find when they all vibrate together again:
- Time for LCM: 24 intervals = 24 seconds
However, since the question states the intervals are in seconds, we need to divide the LCM by the total number of vibrations:
- Final Time Interval: 24 seconds corresponds to 1/2 second of actual vibration time.
Conclusion
Thus, the time interval before all three instruments vibrate together again is 1/2 second (option A).
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