What will be the acute angle between the hour hand and the minute hand at 6:25 p.m.?

Options

A. `30^@`
B. `40.5^@`
C. `42.5^@`
D. `35.5^@`

Answer

To find the acute angle between the hour hand and the minute hand at 6:25 p.m., we can follow these steps:

### Step 1: Determine the position of the minute hand
The minute hand moves 360 degrees in 60 minutes. Therefore, for each minute, the minute hand moves:

\[ \text{Degrees per minute} = \frac{360 \text{ degrees}}{60 \text{ minutes}} = 6 \text{ degrees per minute} \]

At 25 minutes, the position of the minute hand is:

\[ \text{Position of minute hand} = 25 \text{ minutes} \times 6 \text{ degrees/minute} = 150 \text{ degrees} \]

### Step 2: Determine the position of the hour hand
The hour hand moves 360 degrees in 12 hours. Therefore, for each hour, the hour hand moves:

\[ \text{Degrees per hour} = \frac{360 \text{ degrees}}{12 \text{ hours}} = 30 \text{ degrees per hour} \]

At 6:00, the hour hand is at:

\[ \text{Position at 6:00} = 6 \text{ hours} \times 30 \text{ degrees/hour} = 180 \text{ degrees} \]

Since the hour hand also moves as the minutes pass, we need to calculate how far it has moved in the additional 25 minutes:

\[ \text{Degrees moved by hour hand in 25 minutes} = \frac{25 \text{ minutes}}{60 \text{ minutes}} \times 30 \text{ degrees} = 12.5 \text{ degrees} \]

Thus, the position of the hour hand at 6:25 is:

\[ \text{Position of hour hand} = 180 \text{ degrees} + 12.5 \text{ degrees} = 192.5 \text{ degrees} \]

### Step 3: Calculate the angle between the hour hand and the minute hand
Now we can find the angle between the two hands:

\[ \text{Angle} = |\text{Position of hour hand} - \text{Position of minute hand}| \]

Substituting the values we found:

\[ \text{Angle} = |192.5 \text{ degrees} - 150 \text{ degrees}| = 42.5 \text{ degrees} \]

### Step 4: Determine if the angle is acute
Since 42.5 degrees is less than 90 degrees, it is indeed an acute angle.

### Final Answer
The acute angle between the hour hand and the minute hand at 6:25 p.m. is **42.5 degrees**.

---