The hour hand of a clock is `6 cm` long. The magnitude of the displacement of the tip of hour hand between `1:00` pm to `9:00` pm is `:`

Options

A. `6cm`
B. `6sqrt(3)cm`
C. `4picm`
D. `3sqrt(3)cm`

Explanation

To find the magnitude of the displacement of the tip of the hour hand of a clock from 1:00 PM to 9:00 PM, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Clock Position**:
- The hour hand of the clock is 6 cm long.
- At 1:00 PM, the hour hand points to the 1 on the clock.
- At 9:00 PM, the hour hand points to the 9 on the clock.

2. **Visualizing the Displacement**:
- The displacement we need to calculate is the straight-line distance between the positions of the hour hand at 1:00 PM and 9:00 PM.
- This forms a triangle where the center of the clock is one vertex, and the tips of the hour hand at 1:00 PM and 9:00 PM are the other two vertices.

3. **Calculating the Angle**:
- The angle between the hour hand at 1:00 PM and 9:00 PM can be calculated.
- From 1 to 9, the hour hand moves through 8 hours.
- Each hour represents 30 degrees (360 degrees/12 hours), so the angle covered is:
\[ \text{Angle} = 8 \times 30^\circ = 240^\circ \]

4. **Using the Law of Cosines**:
- We can use the Law of Cosines to find the displacement (let's denote it as \(d\)):
\[ d^2 = a^2 + b^2 - 2ab \cos(C) \]
where \(a\) and \(b\) are the lengths of the hour hand (6 cm each), and \(C\) is the angle between them (240 degrees).
- Substituting the values:
\[ d^2 = 6^2 + 6^2 - 2 \cdot 6 \cdot 6 \cdot \cos(240^\circ) \]

5. **Calculating Cosine**:
- The cosine of 240 degrees is:
\[ \cos(240^\circ) = -\frac{1}{2} \]
- Thus, substituting this into the equation:
\[ d^2 = 36 + 36 - 2 \cdot 6 \cdot 6 \cdot \left(-\frac{1}{2}\right) \]
\[ d^2 = 36 + 36 + 36 = 108 \]

6. **Finding the Displacement**:
- Taking the square root to find \(d\):
\[ d = \sqrt{108} = 6\sqrt{3} \text{ cm} \]

### Final Answer:
The magnitude of the displacement of the tip of the hour hand between 1:00 PM and 9:00 PM is \(6\sqrt{3} \text{ cm}\).