What is the angle between the hands of a clock when the time is 15 minutes past 6?

Options

A. `97.5^(@)`
B. `87.5^(@)`
C. `77^(@)`
D. none of these

Answer

To find the angle between the hands of a clock when the time is 15 minutes past 6, we can use the formula:

\[ \text{Angle} = \left| \frac{11}{2} m - 30 h \right| \]

where:
- \( m \) is the number of minutes past the hour,
- \( h \) is the hour.

### Step 1: Identify the values of \( m \) and \( h \)
Here, the time is 6:15.
- \( m = 15 \) (minutes)
- \( h = 6 \) (hours)

### Step 2: Substitute the values into the formula
Now we substitute \( m \) and \( h \) into the formula:

\[ \text{Angle} = \left| \frac{11}{2} \times 15 - 30 \times 6 \right| \]

### Step 3: Calculate \( \frac{11}{2} \times 15 \)
Calculating \( \frac{11}{2} \times 15 \):

\[ \frac{11 \times 15}{2} = \frac{165}{2} = 82.5 \]

### Step 4: Calculate \( 30 \times 6 \)
Calculating \( 30 \times 6 \):

\[ 30 \times 6 = 180 \]

### Step 5: Substitute back into the angle formula
Now substitute these values back into the angle formula:

\[ \text{Angle} = \left| 82.5 - 180 \right| \]

### Step 6: Calculate the absolute difference
Calculating the absolute difference:

\[ \text{Angle} = \left| -97.5 \right| = 97.5 \]

### Final Result
Thus, the angle between the hands of the clock when the time is 15 minutes past 6 is:

\[ \text{Angle} = 97.5^\circ \]