The angle between the minute and hour hands of a clock at 8:30 is

Options

A. `75^@`
B. `80^@`
C. `105^@`
D. `60^@`

Answer

To find the angle between the minute and hour hands of a clock at 8:30, we can follow these steps:

### Step 1: Calculate the angle traced by the hour hand
The hour hand completes a full circle (360 degrees) in 12 hours. Therefore, the angle traced by the hour hand in one hour is:
\[ \text{Angle per hour} = \frac{360 \text{ degrees}}{12 \text{ hours}} = 30 \text{ degrees per hour} \]
At 8:30, the hour hand has moved for 8 hours and an additional half hour. Thus, the total angle traced by the hour hand is:
\[ \text{Angle by hour hand} = 30 \text{ degrees/hour} \times (8 + 0.5) \text{ hours} = 30 \times 8.5 = 255 \text{ degrees} \]

### Step 2: Calculate the angle traced by the minute hand
The minute hand completes a full circle (360 degrees) in 60 minutes. Therefore, the angle traced by the minute hand in one minute is:
\[ \text{Angle per minute} = \frac{360 \text{ degrees}}{60 \text{ minutes}} = 6 \text{ degrees per minute} \]
At 8:30, the minute hand has moved for 30 minutes. Thus, the total angle traced by the minute hand is:
\[ \text{Angle by minute hand} = 6 \text{ degrees/minute} \times 30 \text{ minutes} = 180 \text{ degrees} \]

### Step 3: Calculate the angle between the hour and minute hands
To find the angle between the hour and minute hands, we take the absolute difference between the angles calculated in steps 1 and 2:
\[ \text{Angle between hands} = |255 \text{ degrees} - 180 \text{ degrees}| = 75 \text{ degrees} \]

### Final Answer
The angle between the minute and hour hands of a clock at 8:30 is **75 degrees**.

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