The height of a house subtends a right angle at the opposite street light. The angle of elevation of light from the base of the house is `60^(@)`. If the width of the road be 6 meters, then the height of the house is

Options

A. `8sqrt(3)` m
B. 8 m
C. 6 m
D. `6sqrt(3)`m

Answer

To find the height of the house given the angle of elevation and the width of the road, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- Let the height of the house be \( h \).
- The street light is at point \( L \) and the base of the house is at point \( B \).
- The distance from the house to the street light (width of the road) is given as \( 6 \) meters.
- The angle of elevation from point \( B \) to point \( L \) is \( 60^\circ \).

2. **Identify the Right Triangle**:
- In triangle \( BCD \) (where \( C \) is the point directly below the street light), we can use the tangent of the angle of elevation:
\[ \tan(60^\circ) = \frac{CD}{BD} \]
Here, \( CD \) is the height of the street light (which we will denote as \( x \)), and \( BD \) is the width of the road, which is \( 6 \) meters.

3. **Apply the Tangent Function**:
- We know that \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{x}{6} \]
- Rearranging gives:
\[ x = 6 \sqrt{3} \]

4. **Determine the Height of the House**:
- Now, we need to consider triangle \( ACE \) where \( A \) is the top of the house, \( C \) is the base of the street light, and \( E \) is the point directly below the height of the house.
- The angle \( ACB \) is \( 90^\circ \) and the angle \( CAB \) is \( 30^\circ \) (since \( 60^\circ + 30^\circ = 90^\circ \)).
- In triangle \( ACE \):
\[ \tan(30^\circ) = \frac{AE}{CE} \]
- Here, \( AE = h - x \) and \( CE = 6 \):
\[ \frac{1}{\sqrt{3}} = \frac{h - x}{6} \]
- Rearranging gives:
\[ h - x = \frac{6}{\sqrt{3}} \]

5. **Substituting for \( x \)**:
- Substitute \( x = 6\sqrt{3} \) into the equation:
\[ h - 6\sqrt{3} = \frac{6}{\sqrt{3}} \]
- Simplifying \( \frac{6}{\sqrt{3}} \) gives \( 2\sqrt{3} \):
\[ h - 6\sqrt{3} = 2\sqrt{3} \]

6. **Solve for \( h \)**:
- Rearranging gives:
\[ h = 6\sqrt{3} + 2\sqrt{3} = 8\sqrt{3} \]

### Final Answer:
The height of the house is \( h = 8\sqrt{3} \) meters.