What are the possible times when a clock shows `35^(@)` angle between two hands between 3 pm and 4 pm?

Options

A. A. 20 min 25 sec
B. B. 25 min 20 sec
C. C. 22 min 43 sec
D. D. none of these

Answer

To solve the problem of finding the possible times when a clock shows a 35-degree angle between the two hands between 3 PM and 4 PM, we can use the formula for the angle between the hour and minute hands of a clock:

\[ \theta = \left| \frac{11y - 60x}{2} \right| \]

Where:
- \( \theta \) is the angle between the two hands,
- \( y \) is the number of minutes past the hour,
- \( x \) is the hour.

### Step-by-Step Solution:

1. **Identify the values for \( x \) and \( \theta \)**:
- Since we are looking for the angle between 3 PM and 4 PM, we set \( x = 3 \) (the hour).
- The given angle \( \theta = 35^\circ \).

2. **Substitute the values into the formula**:
\[ 35 = \left| \frac{11y - 60 \cdot 3}{2} \right| \]

3. **Calculate \( 60 \cdot 3 \)**:
\[ 60 \cdot 3 = 180 \]
So the equation becomes:
\[ 35 = \left| \frac{11y - 180}{2} \right| \]

4. **Eliminate the absolute value**:
This gives us two equations to solve:
- \( \frac{11y - 180}{2} = 35 \)
- \( \frac{11y - 180}{2} = -35 \)

5. **Solve the first equation**:
\[ \frac{11y - 180}{2} = 35 \]
Multiply both sides by 2:
\[ 11y - 180 = 70 \]
Add 180 to both sides:
\[ 11y = 250 \]
Divide by 11:
\[ y = \frac{250}{11} \approx 22.727 \text{ minutes} \]

6. **Convert to minutes and seconds**:
- The whole number part is 22 minutes.
- The decimal part \( 0.727 \) minutes can be converted to seconds:
\[ 0.727 \times 60 \approx 43.636 \text{ seconds} \approx 44 \text{ seconds} \]
Thus, the time is approximately **22 minutes and 44 seconds**.

7. **Solve the second equation**:
\[ \frac{11y - 180}{2} = -35 \]
Multiply both sides by 2:
\[ 11y - 180 = -70 \]
Add 180 to both sides:
\[ 11y = 110 \]
Divide by 11:
\[ y = 10 \text{ minutes} \]

### Final Possible Times:
- The possible times when the clock shows a 35-degree angle between the hands are:
- **3:22:44 PM**
- **3:10:00 PM**