A light house has two lights one that flashes every 3 minutes and another that flashes every 3 and a half minute suppose the light flash together at noon what is the first time after 2 p.m. that will they will flash together again
the 1/2 minutes will add up to another 3.5 minutes after 7 3-minute cycles
So, at 2:21 they will both blink again.
Oh, that's a glowing question! Let me shed some light on this.
To find the first time after 2 p.m. when both lights will flash together again, we need to find the least common multiple (LCM) of 3 and 3.5.
The LCM of 3 and 3.5 is 7, so both lights will flash together every 7 minutes.
Now, let's calculate the time after 2 p.m. when they will flash together again.
Between 2 p.m. and 3 p.m., there are 60 minutes.
From 3 p.m. to 4 p.m., there are another 60 minutes, totaling 120 minutes so far.
Continuing this pattern, we'll need to go another 7 minutes past 4 p.m., which brings us to 4:07 p.m.
Therefore, the first time after 2 p.m. that both lights will flash together again is at 4:07 p.m.
To determine the time when the lighthouse lights will flash together again, we need to find the least common multiple (LCM) of the flashing intervals of each light.
The first light flashes every 3 minutes, so its intervals can be represented by the sequence: 3, 6, 9, 12, 15, ...
The second light flashes every 3.5 minutes, so its intervals can be represented by the sequence: 3.5, 7, 10.5, 14, 17.5, ...
To find the LCM of these two sequences, we can write out the intervals and look for the first common value:
3, 6, 9, 12, 15, ...
3.5, 7, 10.5, 14, 17.5, ...
We can determine the LCM by finding their common multiples:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
Multiples of 3.5: 3.5, 7, 10.5, 14, 17.5, 21, 24.5, 28, 31.5, 35, ...
Based on this calculation, we can see that the common multiple 21 is the least common multiple. This means that the lights will flash together again every 21 minutes.
Since the lights initially flashed together at noon, we need to calculate the subsequent time when they will flash together again after 2 p.m. (which is 14:00):
To calculate the time interval from 2 p.m. to the next flash, we subtract 14:00 (in minutes) from 21 and then add the result to the current time.
21 - (14 × 60) = 21 - 840 = -819
Since the resulting time is negative, we need to add it to the total minutes in a day (24 hours × 60 minutes):
-819 + (24 × 60) = -819 + 1440 = 621
The calculation yields a positive result, indicating that the next time the lights will flash together again is 621 minutes after 2 p.m.
To convert this back to regular time, we divide the minutes by 60 to get the hours and find the remainder to get the minutes:
621 ÷ 60 = 10 hours and remainder 21 minutes
Therefore, the lights will flash together again at 12:21 a.m. the following day, which is the first time after 2 p.m.
To find the first time after 2 p.m. that the two lights will flash together again, we need to find the least common multiple (LCM) of their flash intervals.
Let's analyze the flash intervals:
The first light flashes every 3 minutes, which can be written as 3/1 minutes.
The second light flashes every 3 and a half minutes, which can be written as 7/2 minutes.
To find the LCM, we can follow these steps:
Step 1: Express the fractions with a common denominator:
3/1 minutes becomes 6/2 minutes.
7/2 minutes remains the same.
Step 2: Find the LCM of the denominators (2, 2):
The LCM of 2 and 2 is 2.
Step 3: Multiply each numerator by the factor required to reach the LCM:
For the first light: (6/2) x 1 = 6/2 minutes.
For the second light: (7/2) x 1 = 7/2 minutes.
So, the flash intervals are 6/2 minutes and 7/2 minutes, respectively.
Step 4: Find the LCM of the flash intervals:
The LCM of 6/2 and 7/2 is 21/2 minutes.
This means that the two lights will flash together every 21/2 minutes.
Now, let's calculate the time from noon to 2 p.m., which is a total of 2 hours:
2 hours x 60 minutes/hour = 120 minutes.
Finally, we add this duration to noon to find the first time after 2 p.m. that the lights will flash together again:
Noon + 120 minutes = 2 p.m.
2 p.m. + 21/2 minutes = 2 p.m. + 10.5 minutes = 2:10.5 p.m.
Therefore, the first time after 2 p.m. that the two lights will flash together again is 2:10.5 p.m.