When it is 10:00, the hour hand on an analog clock is 300 degrees ahead of the minute hand. How many minutes elapse (to three significant digits) before the minute hand next points in the same direction as the hour hand?

The hour hand moves 30 degrees per hour, or 0.5 degrees per minute.

The minutes hand moves 360 degrees per hour, or 6 degrees per minute.
The difference between the two is 5.5 degrees per minute.
To catch up with 300 degrees difference, it takes: 300/5.5 minutes = 600/11 = 54 6/11 minutes.

To solve this problem, let's break it down step by step:

Step 1: Calculate the angular speed of each hand
The minute hand moves 360 degrees in 60 minutes, so its angular speed is:
Minute hand angular speed = 360 degrees / 60 minutes = 6 degrees per minute

The hour hand moves 360 degrees in 12 hours (720 minutes), so its angular speed is:
Hour hand angular speed = 360 degrees / 720 minutes = 0.5 degrees per minute

Step 2: Calculate the initial angle difference between the hour and minute hands
Since the hour hand is 300 degrees ahead of the minute hand at 10:00, the initial angle difference is:
Initial angle difference = 300 degrees

Step 3: Determine the time it takes for the minute hand to "catch up" to the hour hand
To find the time it takes for the minute hand to catch up to the hour hand, we need to find the difference between their angular speeds. Since the minute hand is "gaining" on the hour hand by 6 degrees per minute, the time it takes for it to catch up is:
Catch up time = Initial angle difference / Difference in angular speeds
Catch up time = 300 degrees / (6 degrees per minute - 0.5 degrees per minute)

Step 4: Calculate the total elapsed time in minutes
To find the total elapsed time, we multiply the catch up time by the number of minutes per hour and round to three significant digits:
Total elapsed time = Catch up time in minutes
Total elapsed time = Catch up time * 60 minutes per hour

Following these steps, you should be able to calculate the total elapsed time in minutes.

To find out how many minutes will elapse before the minute hand next points in the same direction as the hour hand, we need to determine how many degrees the minute hand moves per minute and how many degrees the hour hand moves per minute.

In a 12-hour analog clock, the hour hand completes a full revolution (360 degrees) in 12 hours, equivalent to 720 minutes (12 * 60). Therefore, the hour hand moves at a rate of 360 degrees/720 minutes = 0.5 degrees per minute.

The minute hand, on the other hand, completes a full revolution in 60 minutes, equivalent to 360 degrees. Hence, it moves at a rate of 360 degrees/60 minutes = 6 degrees per minute.

Now, we know that when it is 10:00, the hour hand is 300 degrees ahead of the minute hand. This means that the hour hand has moved 300 degrees more than the minute hand. Therefore, using the rates we calculated above, we can set up the equation:

0.5(degrees/minute) * t(minutes) = 300 degrees

Solving for t gives us:

t = 300 degrees / (0.5 degrees/minute) = 600 minutes

Therefore, it will take 600 minutes for the minute hand to catch up with the hour hand again.