At what time between 3 oclock and 4 o clock will the hands of a clock be together

Options

A. `16(3)/(11)` min past 3
B. `14(3)/(11)` min past 3
C. `13(2)/(11)` min past 3
D. `16(4)/(11)` min past 3

Answer

To find the time between 3 o'clock and 4 o'clock when the hands of a clock are together, we can use the formula for the angle between the hour and minute hands:

\[ \theta = |30h - 5.5m| \]

Where:
- \( \theta \) is the angle between the hour and minute hands,
- \( h \) is the hour,
- \( m \) is the minutes.

Since we want the hands to be together, we set \( \theta = 0 \):

1. **Identify the values for h and m**:
- Here, \( h = 3 \) (since we are between 3 and 4 o'clock).
- We need to find \( m \) (the minutes after 3 o'clock).

2. **Set up the equation**:
\[ 0 = |30h - 5.5m| \]
Substituting \( h = 3 \):
\[ 0 = |30 \times 3 - 5.5m| \]
This simplifies to:
\[ 0 = |90 - 5.5m| \]

3. **Remove the absolute value**:
This gives us two equations:
\[ 90 - 5.5m = 0 \quad \text{or} \quad 90 - 5.5m = 0 \]

4. **Solve for m**:
From \( 90 - 5.5m = 0 \):
\[ 5.5m = 90 \]
\[ m = \frac{90}{5.5} = \frac{900}{55} = \frac{180}{11} \approx 16.36 \]

5. **Convert m to minutes and seconds**:
The value \( \frac{180}{11} \) minutes is approximately 16 minutes and \( \frac{180 \mod 11}{11} \times 60 \) seconds.
- \( 180 \mod 11 = 3 \)
- Therefore, \( \frac{3}{11} \times 60 \approx 16.36 \) seconds.

6. **Final time**:
The hands of the clock will be together at approximately 16 minutes and 21.82 seconds after 3 o'clock.

Thus, the time when the hands of the clock are together between 3 o'clock and 4 o'clock is approximately **3:16 and 21.82 seconds**.