A cubicla room is formed with 6 plane mirrors. An insect moves along the diagonal of the floor with uniform speed. The speed of its image in two adjacent walls is `20 sqrt2 cm//s`.
Then the speed of the image formed on the roof is

Options

A. `20 cm//s`
B. 40 cm/s
C. `20 sqrt2 cm//s`
D. `10sqrt2 cm//s`

Answer

To solve the problem, we need to analyze the movement of the insect and the formation of its image in the cubic room formed by mirrors.

### Step-by-step Solution:

1. **Understanding the Setup**:
- The insect is moving along the diagonal of the floor of a cubic room.
- The speed of the insect is denoted as \( V \).
- The speed of the image of the insect in two adjacent walls is given as \( 20\sqrt{2} \, \text{cm/s} \).

2. **Components of Speed**:
- Since the insect is moving along the diagonal, we can break down its speed into components along the axes of the cubic room.
- The diagonal of the floor can be represented in a 2D plane (let's say the x-y plane) where the insect moves from one corner of the room to the opposite corner.

3. **Relating Image Speed to Insect Speed**:
- The speed of the image formed on the adjacent walls can be expressed in terms of the speed of the insect.
- If the speed of the insect is \( V \), then the speed of the image in the direction of each wall can be calculated using the relationship:
\[ \text{Speed of image} = V \cdot \frac{1}{\sqrt{2}} \]
- This is because the diagonal movement creates equal components along both axes.

4. **Setting Up the Equation**:
- We know from the problem that the speed of the image in the two adjacent walls is \( 20\sqrt{2} \, \text{cm/s} \):
\[ V \cdot \frac{1}{\sqrt{2}} = 20\sqrt{2} \]

5. **Solving for \( V \)**:
- To find \( V \), we multiply both sides by \( \sqrt{2} \):
\[ V = 20\sqrt{2} \cdot \sqrt{2} \]
- Simplifying gives:
\[ V = 20 \cdot 2 = 40 \, \text{cm/s} \]

6. **Finding the Speed of the Image on the Roof**:
- The speed of the image formed on the roof will also be equal to the speed of the insect \( V \), because the roof is parallel to the floor and the insect’s motion is uniform.
- Therefore, the speed of the image on the roof is also \( 40 \, \text{cm/s} \).

### Final Answer:
The speed of the image formed on the roof is \( 40 \, \text{cm/s} \).