What is the radian measure of the smaller angle formed by the hands of a clock at 7 o'clock?

Options

A. `(pi)/(2)`
B. `(2pi)/(3)`
C. `(5pi)/(6)`
D. `(7pi)/(6)`

Answer

To find the radian measure of the smaller angle formed by the hands of a clock at 7 o'clock, we can follow these steps:

### Step 1: Understand the positions of the clock hands
At 7 o'clock, the hour hand is on the 7 and the minute hand is on the 12.

### Step 2: Calculate the angle for one hour interval
The entire clock represents a full circle, which is \(2\pi\) radians. Since there are 12 hours on a clock, the angle for each hour interval is:
\[ \text{Angle for one interval} = \frac{2\pi}{12} = \frac{\pi}{6} \]

### Step 3: Determine the number of intervals between 7 and 12
To find the angle between the hour hand at 7 and the minute hand at 12, we need to count the number of hour intervals from 7 to 12:
- From 7 to 8: 1 interval
- From 8 to 9: 2 intervals
- From 9 to 10: 3 intervals
- From 10 to 11: 4 intervals
- From 11 to 12: 5 intervals

Thus, there are 5 intervals between 7 and 12.

### Step 4: Calculate the total angle between the hands
Now, we can calculate the total angle formed by the hour hand and the minute hand:
\[ \text{Total angle} = \text{Number of intervals} \times \text{Angle for one interval} = 5 \times \frac{\pi}{6} = \frac{5\pi}{6} \]

### Step 5: Identify the smaller angle
Since the total angle between the hands is \(\frac{5\pi}{6}\) radians, and this is less than \(\pi\) radians, it is already the smaller angle.

### Final Answer
The radian measure of the smaller angle formed by the hands of a clock at 7 o'clock is:
\[ \frac{5\pi}{6} \text{ radians} \]

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