Between 3 O'clock and 4 O'clock when will the two hands make `36^(@)` angle between them: when hour-hand is ahead of minute-hand.

Answer

To solve the problem of when the hour hand is ahead of the minute hand and they form a 36-degree angle between 3 o'clock and 4 o'clock, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Clock**:
- A clock makes a full circle of 360 degrees in 60 minutes.
- Therefore, the angle made by the minute hand in one minute is:
\[ \text{Angle per minute} = \frac{360 \text{ degrees}}{60 \text{ minutes}} = 6 \text{ degrees per minute} \]

2. **Finding the Hour Hand Movement**:
- The hour hand moves 30 degrees for every hour (360 degrees / 12 hours).
- In one minute, the hour hand moves:
\[ \text{Hour hand movement per minute} = \frac{30 \text{ degrees}}{60 \text{ minutes}} = 0.5 \text{ degrees per minute} \]

3. **Position of the Hands at 3:00**:
- At 3:00, the hour hand is at 90 degrees (3 hours × 30 degrees).
- The minute hand is at 0 degrees.

4. **Setting Up the Equation**:
- Let \( t \) be the time in minutes after 3:00.
- The position of the minute hand after \( t \) minutes is:
\[ \text{Minute hand position} = 6t \text{ degrees} \]
- The position of the hour hand after \( t \) minutes is:
\[ \text{Hour hand position} = 90 + 0.5t \text{ degrees} \]

5. **Finding the Angle Between the Hands**:
- The angle between the hour hand and the minute hand can be calculated as:
\[ \text{Angle} = |(90 + 0.5t) - (6t)| = |90 - 5.5t| \]
- We want this angle to be 36 degrees:
\[ |90 - 5.5t| = 36 \]

6. **Solving the Absolute Value Equation**:
- This gives us two cases to solve:
- Case 1: \( 90 - 5.5t = 36 \)
- Case 2: \( 90 - 5.5t = -36 \)

**Case 1**:
\[ 90 - 5.5t = 36 \implies 5.5t = 90 - 36 \implies 5.5t = 54 \implies t = \frac{54}{5.5} \approx 9.82 \text{ minutes} \]

**Case 2**:
\[ 90 - 5.5t = -36 \implies 5.5t = 90 + 36 \implies 5.5t = 126 \implies t = \frac{126}{5.5} \approx 22.91 \text{ minutes} \]

7. **Final Answer**:
- The two times when the angle is 36 degrees between the hour and minute hands, with the hour hand ahead, are approximately:
- 9.82 minutes past 3:00 (which is around 3:10)
- 22.91 minutes past 3:00 (which is around 3:23)