At what time between 5 o clock and 6 o clock in the morning will the both hands of a clock be art right angle

Options

A. `43(7)/(11)` min past 5 and `1(10)/(11)` min past 6
B. `43(7)/(11)` min past 5 and `10(10)/(11)` min past 5
C. `42(7)/(11)` min past 5 and `11(10)/(11)` min past 5
D. `42(7)/(11)` min past 5 and `1(10)/(11)` min past 6

Answer

To find the time between 5 o'clock and 6 o'clock when the hands of a clock are at right angles, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Clock Positions**:
- The hour hand moves at a rate of 30 degrees per hour (360 degrees/12 hours).
- The minute hand moves at a rate of 6 degrees per minute (360 degrees/60 minutes).

2. **Position of the Hour Hand at 5:00**:
- At 5:00, the hour hand is at 5 × 30 = 150 degrees from the 12 o'clock position.

3. **Finding Right Angles**:
- The hands of the clock are at right angles when the angle between them is 90 degrees or 270 degrees.
- We will calculate the times when the angle is 90 degrees.

4. **Setting Up the Equation**:
- Let the number of minutes after 5:00 be \( m \).
- The position of the minute hand after \( m \) minutes is \( 6m \) degrees.
- The position of the hour hand after \( m \) minutes is \( 150 + 0.5m \) degrees (since it moves 0.5 degrees per minute).

5. **Equation for Right Angles**:
- For the first case (90 degrees apart):
\[ |(150 + 0.5m) - (6m)| = 90 \]
- This simplifies to:
\[ |150 - 5.5m| = 90 \]

6. **Solving the Equation**:
- This gives us two cases to solve:
- Case 1: \( 150 - 5.5m = 90 \)
- Case 2: \( 150 - 5.5m = -90 \)

- **Case 1**:
\[ 150 - 5.5m = 90 \implies 5.5m = 60 \implies m = \frac{60}{5.5} \approx 10.91 \text{ minutes} \]

- **Case 2**:
\[ 150 - 5.5m = -90 \implies 5.5m = 240 \implies m = \frac{240}{5.5} \approx 43.64 \text{ minutes} \]

7. **Converting Minutes to Time**:
- For Case 1: \( 5:10.91 \) (approximately 5:11)
- For Case 2: \( 5:43.64 \) (approximately 5:44)

### Final Answer:
The hands of the clock will be at right angles at approximately **5:11 AM** and **5:44 AM**.