at how many minutes after 3pm will the minute hand of a clock overtake the hour hand?
3:16.36
x=(15+(x/12))
11x=180
x=16.36
Time=3:16:21.81
Well, it seems like those clock hands are always trying to one-up each other! To answer your question, let's analyze the situation.
In a regular clock, the hour hand moves 12 times slower than the minute hand (since an hour has 60 minutes and a full circle on the clock has 12 hour markings).
So, for every hour that passes, the minute hand gains a lead over the hour hand equal to 55 minutes (60 minutes - 5 minutes that the hour hand has already covered).
Since there are 12 hours on a clock face, and we want to find out how long it takes for the minute hand to overtake the hour hand after 3pm, we can calculate it like this:
Minutes = (number of hours) x (lead minutes)
Minutes = (12 hours) x (55 minutes)
Minutes = 660 minutes
So, the minute hand will overtake the hour hand 660 minutes after 3pm. Therefore, it will happen at 3:00 pm + 660 minutes, which gives us 10:00 pm.
But keep in mind, this is assuming that we're talking about a traditional analog clock. If we're talking about a digital clock, well, the minute hand doesn't quite have the opportunity to show off in the same way. So, let's assume we're talking about an analog clock.
To determine at what time the minute hand of a clock will overtake the hour hand, we need to calculate the time difference in minutes between 3pm and when the minute hand catches up to the hour hand.
Here's how you can solve this:
1. Determine the positions of the hour and minute hands at 3pm:
- At 3pm, the hour hand points directly at the 3.
- The minute hand points at the 12.
2. Calculate the hour's position in minutes:
- Since each hour on a clock corresponds to 30 degrees (360 degrees in a full circle divided by 12 hours), we can say that each hour represents 30 minutes on the clock face.
- At 3pm, the hour hand is at the 3, which implies it is 15 minutes past the hour.
3. Calculate the speed difference between the minute and hour hands:
- The minute hand moves 12 times faster than the hour hand because there are 12 equal divisions (hours) on a clock face.
- Therefore, the minute hand gains on the hour hand at a rate of 11 minutes per hour (12 minutes - 1 minute).
4. Calculate the time it takes for the minute hand to overtake the hour hand:
- Since the minute hand gains 11 minutes per hour on the hour hand, it will take 15 / 11 hours for the minute hand to catch up to the hour hand from 3pm.
- Converting 15 / 11 hours into minutes: (15 / 11) * 60 = approximately 81.82 minutes.
Therefore, the minute hand will overtake the hour hand approximately 81.82 minutes after 3pm.