Find the angle between the hour hand and the minute hand of a clock when the time is 4:10

Options

A. `65^(@)`
B. `35^(@)`
C. `125^(@)`
D. `60^(@)`

Answer

To find the angle between the hour hand and the minute hand of a clock at 4:10, we can follow these steps:

### Step 1: Calculate the position of the hour hand
The hour hand moves 30 degrees for each hour (since 360 degrees / 12 hours = 30 degrees per hour).

At 4:00, the hour hand is at:
\[ 4 \times 30 = 120 \text{ degrees} \]

Since the hour hand also moves as the minutes pass, we need to account for the additional 10 minutes. The hour hand moves 0.5 degrees for each minute (since 30 degrees / 60 minutes = 0.5 degrees per minute).

For 10 minutes, the hour hand moves:
\[ 10 \times 0.5 = 5 \text{ degrees} \]

Thus, at 4:10, the position of the hour hand is:
\[ 120 + 5 = 125 \text{ degrees} \]

### Step 2: Calculate the position of the minute hand
The minute hand moves 6 degrees for each minute (since 360 degrees / 60 minutes = 6 degrees per minute).

At 10 minutes past the hour, the position of the minute hand is:
\[ 10 \times 6 = 60 \text{ degrees} \]

### Step 3: Calculate the angle between the hour hand and the minute hand
Now, we find the difference between the positions of the hour hand and the minute hand:
\[ |125 - 60| = 65 \text{ degrees} \]

### Conclusion
The angle between the hour hand and the minute hand at 4:10 is:
\[ \text{Angle} = 65 \text{ degrees} \]

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