At what time after 3:10 am, the acute angle made by the minute and hour-hand is double to that of at 3:10 am, for the first time?

Options

A. 4 h 43 min
B. 3 h 48 min
C. 3h 320/11 min
D. none of the these

Answer

To solve the problem of finding the time after 3:10 am when the acute angle made by the minute and hour hands is double that of the angle at 3:10 am, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the angle at 3:10 am**:
We use the formula for the angle between the hour hand and minute hand:
\[ \text{Angle} = |30h - \frac{11}{2}m| \]
where \(h\) is the hour and \(m\) is the minutes.

For 3:10 am:
- \(h = 3\)
- \(m = 10\)

Plugging in the values:
\[ \text{Angle} = |30 \times 3 - \frac{11}{2} \times 10| = |90 - 55| = |35| \text{ degrees} \]

2. **Determine the double of the angle**:
We need to find when the angle is double that of 3:10 am:
\[ 2 \times 35 = 70 \text{ degrees} \]

3. **Set up the equation for the angle to be 70 degrees**:
We need to find the time after 3:10 am when the angle is 70 degrees. Let \(m\) be the minutes after 3:10 am. The time can be expressed as \(3:10 + m\) minutes.

The new hour and minute values will be:
- Hour \(h = 3\)
- Minutes \(m + 10\)

The angle equation becomes:
\[ |30 \times 3 - \frac{11}{2}(m + 10)| = 70 \]

4. **Solve the equation**:
This leads to two cases due to the absolute value.

**Case 1**:
\[ 90 - \frac{11}{2}(m + 10) = 70 \]
Simplifying:
\[ 90 - 70 = \frac{11}{2}(m + 10) \]
\[ 20 = \frac{11}{2}(m + 10) \]
\[ 40 = 11(m + 10) \]
\[ 40 = 11m + 110 \]
\[ 11m = 40 - 110 \]
\[ 11m = -70 \quad \text{(not possible)} \]

**Case 2**:
\[ 90 - \frac{11}{2}(m + 10) = -70 \]
Simplifying:
\[ 90 + 70 = \frac{11}{2}(m + 10) \]
\[ 160 = \frac{11}{2}(m + 10) \]
\[ 320 = 11(m + 10) \]
\[ 320 = 11m + 110 \]
\[ 11m = 320 - 110 \]
\[ 11m = 210 \]
\[ m = \frac{210}{11} \approx 19.09 \]

5. **Calculate the time**:
The time after 3:10 am when the angle is 70 degrees for the first time is:
\[ 3:10 + 19.09 \text{ minutes} \approx 3:29.09 \text{ am} \]

### Final Answer:
The time after 3:10 am when the acute angle made by the minute and hour hands is double that of the angle at 3:10 am is approximately **3:29 am**.