At what time are the hands of clock together between 7 pm and 8 pm?

Options

A. `7:45:54`
B. `7:36:27`
C. `7:37:49`
D. 7:38:18

Answer

To find the time at which the hands of the clock are together between 7 PM and 8 PM, we can use the formula for the position of the hour and minute hands.

### Step-by-Step Solution:

1. **Understanding the Formula**:
The formula to find the time (in minutes) when the hands of the clock are together is given by:
\[ \theta = 30H - \frac{11}{2}M \]
where \( H \) is the hour and \( M \) is the minutes past the hour. When the hands are together, \( \theta = 0 \).

2. **Setting Up the Equation**:
For the time between 7 PM and 8 PM, we have:
- \( H = 7 \)
- The equation becomes:
\[ 0 = 30 \times 7 - \frac{11}{2}M \]

3. **Calculating the Constant**:
Calculate \( 30 \times 7 \):
\[ 30 \times 7 = 210 \]
So, the equation now is:
\[ 0 = 210 - \frac{11}{2}M \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{11}{2}M = 210 \]

5. **Solving for M**:
Multiply both sides by \( \frac{2}{11} \):
\[ M = \frac{210 \times 2}{11} = \frac{420}{11} \approx 38.18 \]

6. **Interpreting the Result**:
The value \( M \approx 38.18 \) means that the hands of the clock are together approximately 38 minutes and 11 seconds after 7 PM.

7. **Final Answer**:
Therefore, the hands of the clock are together at approximately **7:38:11 PM**.