When do the two hands of a clock of just after 3 pm make `30^(@)` angle between them?

Options

A. `3:15:00`
B. `3:10:54`
C. `3:01:59`
D. none of these

Answer

To find the time just after 3 PM when the angle between the hour hand and the minute hand of a clock is 30 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Position of the Hands at 3 PM**:
- At 3 PM, the hour hand is at 90 degrees (3 hours × 30 degrees per hour).
- The minute hand is at 0 degrees (at the top of the clock).

2. **Formula for the Angle Between the Hands**:
- The formula to calculate the angle (A) between the hour hand and the minute hand at any given time is:
\[ A = |30H - 5.5M| \]
where \(H\) is the hour and \(M\) is the minutes past the hour.

3. **Set Up the Equation for 30 Degrees**:
- We want the angle \(A\) to be 30 degrees. Therefore, we set up the equation:
\[ |30 \times 3 - 5.5M| = 30 \]
- This simplifies to:
\[ |90 - 5.5M| = 30 \]

4. **Solve the Absolute Value Equation**:
- This gives us two cases to solve:
- Case 1: \(90 - 5.5M = 30\)
- Case 2: \(90 - 5.5M = -30\)

5. **Case 1**: Solve \(90 - 5.5M = 30\):
- Rearranging gives:
\[ 5.5M = 90 - 30 \]
\[ 5.5M = 60 \]
\[ M = \frac{60}{5.5} \approx 10.91 \text{ minutes} \]

6. **Case 2**: Solve \(90 - 5.5M = -30\):
- Rearranging gives:
\[ 5.5M = 90 + 30 \]
\[ 5.5M = 120 \]
\[ M = \frac{120}{5.5} \approx 21.82 \text{ minutes} \]

7. **Convert Minutes to Seconds**:
- For Case 1:
- \(10.91\) minutes is approximately \(10\) minutes and \(54.6\) seconds.
- For Case 2:
- \(21.82\) minutes is approximately \(21\) minutes and \(49.2\) seconds.

8. **Final Times**:
- Therefore, the two times just after 3 PM when the angle between the hands is 30 degrees are:
- **3:10:55** (approximately)
- **3:21:49** (approximately)

### Conclusion:
The two times just after 3 PM when the angle between the clock hands is 30 degrees are approximately **3:10:55** and **3:21:49**.