Find the angle in radians between the hands of a clock at 7: 20 PM

Answer

To find the angle in radians between the hands of a clock at 7:20 PM, we can follow these steps:

### Step 1: Understand the positions of the hands
At 7:20 PM:
- The minute hand points at the 4 (20 minutes).
- The hour hand is slightly past the 7.

### Step 2: Calculate the angle between hour marks
The clock is divided into 12 hours, and the total angle of the clock is 360 degrees. Therefore, the angle between each hour mark is:
\[ \text{Angle per hour} = \frac{360 \text{ degrees}}{12} = 30 \text{ degrees} \]

### Step 3: Calculate the angle between 7 and 4
From 7 to 4, there are 3 hour marks (7 to 8, 8 to 9, and 9 to 10). Thus, the angle from 7 to 4 is:
\[ \text{Angle from 7 to 4} = 30 \text{ degrees} \times 3 = 90 \text{ degrees} \]

### Step 4: Calculate the additional angle for the hour hand
The hour hand moves as time progresses. In 20 minutes, the hour hand moves further. Since 20 minutes is \(\frac{1}{3}\) of an hour, we can find how much the hour hand moves in that time:
\[ \text{Angle moved by hour hand in 20 minutes} = \frac{1}{3} \text{ hour} \times 30 \text{ degrees/hour} = 10 \text{ degrees} \]

### Step 5: Calculate the total angle between the hands
Now, we can add the angle from 7 to 4 (90 degrees) and the additional movement of the hour hand (10 degrees):
\[ \text{Total angle} = 90 \text{ degrees} + 10 \text{ degrees} = 100 \text{ degrees} \]

### Step 6: Convert the angle to radians
To convert degrees to radians, we use the conversion factor \(\frac{\pi \text{ radians}}{180 \text{ degrees}}\):
\[ \text{Angle in radians} = 100 \text{ degrees} \times \frac{\pi}{180} = \frac{100\pi}{180} = \frac{5\pi}{9} \text{ radians} \]

### Final Answer
The angle in radians between the hands of the clock at 7:20 PM is:
\[ \frac{5\pi}{9} \text{ radians} \]