A vertical lamp-post, 6m high, stands at a distance of 2 m from a wall, 4 m high . A 1.5 m tall man starts to walk away from the wall on the other side of the wall, in line with the lamp-post the maximum distance to which the man can walk remaining in the shadow is

Options

A. `5/2`m
B. `3/2`m
C. 4m
D. none of these

Answer

To solve the problem step by step, we will analyze the situation involving the lamp-post, the wall, and the man walking away from the wall.

### Step 1: Understand the Setup
We have:
- A lamp-post of height 6 m.
- A wall of height 4 m, located 2 m away from the lamp-post.
- A man of height 1.5 m who walks away from the wall.

### Step 2: Draw a Diagram
Create a diagram to visualize the situation:
- Label the lamp-post as point A (6 m high).
- Label the wall as point B (4 m high), which is 2 m away from point A.
- The man will be walking away from the wall, so label his position as point C (1.5 m high).

### Step 3: Identify Similar Triangles
To find how far the man can walk while remaining in the shadow, we will use similar triangles:
- Triangle ACD (where D is the point where the shadow of the lamp-post meets the ground).
- Triangle BEC (where E is the point where the shadow of the wall meets the ground).

### Step 4: Set Up Ratios Using Similar Triangles
From the similar triangles, we can set up the following ratios:
1. For triangle ACD and triangle BEC:
\[ \frac{AC}{BC} = \frac{AD}{BE} \]
Where:
- AC = height of the lamp-post = 6 m
- BC = height of the wall = 4 m
- AD = distance from the lamp-post to the point where the shadow meets the ground (unknown)
- BE = distance from the wall to the point where the shadow meets the ground (unknown)

### Step 5: Calculate Distances
Let \( x \) be the distance the man can walk away from the wall while remaining in the shadow. The distance from the wall to the man will be \( 2 + x \).

Using the similar triangles:
\[ \frac{6}{x + 2} = \frac{4}{x} \]

Cross-multiplying gives:
\[ 6x = 4(x + 2) \]
Expanding and simplifying:
\[ 6x = 4x + 8 \\ 2x = 8 \\ x = 4 \]

### Step 6: Conclusion
The maximum distance the man can walk away from the wall while remaining in the shadow is:
\[ \text{Distance} = 4 \text{ m} \]

### Final Answer
The maximum distance to which the man can walk remaining in the shadow is **4 meters**.

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