Through how many radians do the minute hand move between 1:00 p.m. and 6:45 p.m. (on the same day)? 

To find out how many radians the minute hand moves between 1:00 p.m. and 6:45 p.m., we can break the problem down as follows:

### Step 1: Determine the total time elapsed
- The time from 1:00 p.m. to 6:00 p.m. is 5 hours.
- From 6:00 p.m. to 6:45 p.m. is 45 minutes.

So, the total time elapsed is 5 hours and 45 minutes.

### Step 2: Convert the total time to minutes
- 5 hours = \(5 \times 60 = 300\) minutes.
- 45 minutes is already given.

Thus, the total time elapsed in minutes is \(300 + 45 = 345\) minutes.

### Step 3: Calculate the number of radians the minute hand moves per minute
The minute hand completes one full revolution (which is \(2\pi\) radians) in 60 minutes. Therefore, the number of radians per minute is:

\[
\text{Radians per minute} = \frac{2\pi}{60} = \frac{\pi}{30} \text{ radians}
\]

### Step 4: Calculate the total radians moved
Now, multiply the number of radians per minute by the total time in minutes:

\[
\text{Total radians} = 345 \times \frac{\pi}{30} = \frac{345\pi}{30} = \frac{23\pi}{2} \text{ radians}
\]

Thus, the minute hand moves through \(\frac{23\pi}{2}\) radians between 1:00 p.m. and 6:45 p.m.

Would you like any further details or have any questions?

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Here are 5 related questions to consider:
1. How many radians does the hour hand move between 1:00 p.m. and 6:45 p.m.?
2. How far (in degrees) does the minute hand move between 1:00 p.m. and 6:45 p.m.?
3. What is the angle in radians between the minute hand and the hour hand at 6:45 p.m.?
4. If the time was 1:00 p.m. to 8:15 p.m., how many radians would the minute hand move?
5. How many radians does the minute hand move in 24 hours?

**Tip:** Remember that one full revolution of the minute hand equals \(2\pi\) radians, so knowing the fraction of an hour can help you quickly determine the angle in radians.

Discover how many radians the minute hand moves between 1:00 p.m. and 6:45 p.m. on the same day. This detailed solution includes the conversion of time to minutes, calculation of radians per minute, and multiplication to find the total radians moved. Suitable for advanced high school students and anyone interested in trigonometry and circular motion.

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