At what angle the hands of a clock are inclined at 15 minutes past 5? `58 1/2o` (b) `64o` (c) `67 1/2o` (d) `72 1/2o`

Options

A. `67 1/2^@`
B. `37 1/2^@
C. `52 1/2^@`
D. `22 1/2^@`

Explanation

To find the angle between the hour and minute hands of a clock at 15 minutes past 5, we can follow these steps:

### Step 1: Calculate the position of the hour hand
The hour hand moves at a rate of 30 degrees for each hour. At 5:00, the hour hand is at:
\[ 5 \times 30 = 150 \text{ degrees} \]
Since it is 15 minutes past 5, the hour hand will have moved further. The hour hand moves 0.5 degrees for each minute (30 degrees per hour divided by 60 minutes):
\[ \text{Movement of hour hand in 15 minutes} = 15 \times 0.5 = 7.5 \text{ degrees} \]
Thus, the position of the hour hand at 5:15 is:
\[ 150 + 7.5 = 157.5 \text{ degrees} \]

### Step 2: Calculate the position of the minute hand
The minute hand moves at a rate of 6 degrees for each minute. At 15 minutes past the hour, the minute hand is at:
\[ 15 \times 6 = 90 \text{ degrees} \]

### Step 3: Calculate the angle between the hour and minute hands
Now, we can find the angle between the hour and minute hands by taking the absolute difference between their positions:
\[ \text{Angle} = |157.5 - 90| = 67.5 \text{ degrees} \]

### Conclusion
The angle between the hands of the clock at 15 minutes past 5 is:
\[ \boxed{67.5 \text{ degrees}} \]