How long after 7:30 are a clock's minute hand and hour hand 50 degrees apart? The answer is 17.3 minutes, but I don't know how to get it. Help!
I takes a minute hand 60 minutes to go 360 degrees or 6 degrees per minute. It takes an hour hand 12*60=720 minutes to go 360 degrees or .5 per minute. At 7:30, the minute hand is pointing exactly on the 6 and the hour hand is half-way between the 7 and the 8; an angle of 45 degrees. Since the minute hand is behind the hour hand, lets make the angle -45 degrees.
Ok, solve for minute m where -45 + 6m - .5m = 50.
I get m=17.25 which rounds to 17.3 minutes.
To find out how long after 7:30 the clock's minute hand and hour hand are 50 degrees apart, we need to use some basic trigonometry.
First, let's determine how far the minute hand moves in one minute. The minute hand completes a full rotation (360 degrees) in 60 minutes. So, in one minute, it moves 360 degrees divided by 60 minutes, which is 6 degrees.
Next, let's determine how far the hour hand moves in one minute. The hour hand completes a full rotation (360 degrees) in 12 hours, which is 720 minutes. So, in one minute, it moves 360 degrees divided by 720 minutes, which is 0.5 degrees.
Now, let's calculate the initial angle between the minute and hour hand at 7:30. At 7:30, the hour hand is halfway between 7 and 8, so it has moved half of its distance for that hour. We can calculate the angle it has moved as 0.5 degrees per minute multiplied by 30 minutes, which is 15 degrees.
Since there are two scenarios where the minute and hour hand can be 50 degrees apart (moving clockwise or counterclockwise), we would need to solve two equations.
1. For the minute hand moving clockwise:
Let x be the number of minutes after 7:30.
The equation would be: 6x - 15 = 50.
2. For the minute hand moving counterclockwise:
Let y be the number of minutes after 7:30.
The equation would be: 15 - 6y = 50.
Solving these two equations gives us the solutions:
For the minute hand moving clockwise, x = 10.83 minutes.
For the minute hand moving counterclockwise, y = 9.17 minutes.
Since the question asks for the time after 7:30, we need to consider the counterclockwise scenario. Hence, the answer is approximately 9.17 minutes after 7:30, which, rounded to one decimal place, is 9.2 minutes.
Therefore, the correct answer is approximately 9.2 minutes, not 17.3 minutes.