A street light bulb is fixed on a pole 6 m above the level of the street. If a women of height 1.5 m casts a shadow of 3 m, then find how far she is away from the base of the pole.

Options

A. `8`m
B. `7`m
C. `6`m
D. `9`m

Explanation

To solve the problem, we will use the concept of similar triangles. Let's break down the solution step by step.

### Step 1: Understand the Setup
- We have a pole of height 6 m (point A).
- A woman of height 1.5 m (point C) is standing on the ground and casts a shadow of 3 m (point D).
- We need to find the distance (x) from the base of the pole (point B) to the woman (point C).

### Step 2: Draw the Triangles
- Draw triangle ABE where:
- A is the top of the pole (6 m high).
- B is the base of the pole.
- E is the tip of the shadow of the woman.
- Draw triangle CDE where:
- C is the top of the woman (1.5 m high).
- D is the base of the shadow of the woman.
- E is the tip of the shadow.

### Step 3: Identify the Lengths
- The height of the pole (AB) = 6 m.
- The height of the woman (CD) = 1.5 m.
- The length of the shadow of the woman (DE) = 3 m.
- The distance from the base of the pole to the woman (BC) = x.

### Step 4: Set Up the Proportion
Since triangles ABE and CDE are similar (by AA similarity criterion), we can set up the following proportion:

\[ \frac{AB}{AE} = \frac{CD}{DE} \]

Substituting the known values:

\[ \frac{6}{x + 3} = \frac{1.5}{3} \]

### Step 5: Cross Multiply
Cross multiplying gives us:

\[ 6 \cdot 3 = 1.5 \cdot (x + 3) \]

This simplifies to:

\[ 18 = 1.5x + 4.5 \]

### Step 6: Solve for x
Now, isolate x:

\[ 18 - 4.5 = 1.5x \]

\[ 13.5 = 1.5x \]

Dividing both sides by 1.5:

\[ x = \frac{13.5}{1.5} = 9 \]

### Conclusion
The distance from the base of the pole to the woman is **9 meters**.

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