At 3:40 the hour and minute hands of a clock are inclined at

Options

A. `(13pi)/18`
B. `2pi/3`
C. `(5pi)/18`
D. None Of These

Explanation

To find the angle between the hour and minute hands of a clock at 3:40, we can follow these steps:

### Step 1: Calculate the position of the hour hand
The hour hand moves 360 degrees in 12 hours, which means it moves 30 degrees per hour (360 degrees / 12 hours).

At 3:40:
- The hour hand has moved from 3:00 to 3:40.
- In 40 minutes, the hour hand moves an additional \( \frac{40}{60} \) of an hour, which is \( \frac{2}{3} \) of an hour.

So, the total hours from 12:00 to 3:40 is:
\[ 3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3} \text{ hours} \]

Now, we can calculate the angle moved by the hour hand:
\[ \text{Angle of hour hand} = \frac{11}{3} \times 30 = 110 \text{ degrees} \]

### Step 2: Calculate the position of the minute hand
The minute hand moves 360 degrees in 60 minutes, which means it moves 6 degrees per minute (360 degrees / 60 minutes).

At 40 minutes:
\[ \text{Angle of minute hand} = 40 \times 6 = 240 \text{ degrees} \]

### Step 3: Calculate the angle between the hour and minute hands
To find the angle between the two hands, we take the absolute difference between the angles of the hour hand and the minute hand:
\[ \text{Angle between hands} = |240 - 110| = 130 \text{ degrees} \]

### Step 4: Convert the angle from degrees to radians
To convert degrees to radians, we use the conversion factor \( \frac{\pi \text{ radians}}{180 \text{ degrees}} \):
\[ 130 \text{ degrees} = 130 \times \frac{\pi}{180} = \frac{130\pi}{180} = \frac{13\pi}{18} \text{ radians} \]

### Final Answer:
The angle between the hour and minute hands at 3:40 is \( \frac{13\pi}{18} \) radians.

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