The angle of elevation of the top of a building and the top of the chimney on the roof of the building from a point on the ground are x and `45^(@)` respectively .The height of building is h metre .Then the height of the chimney (in metre ) is :

Options

A. `h cot x +h`
B. `h cot x -h`
C. `h tan x- h `
D. `h tan x +h`

Answer

To solve the problem step by step, we will use trigonometric principles related to angles of elevation.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a building of height \( h \) meters.
- There is a chimney on top of the building.
- The angle of elevation to the top of the building from a point on the ground is \( x \) degrees.
- The angle of elevation to the top of the chimney is \( 45^\circ \).

2. **Drawing the Diagram**:
- Draw a vertical line representing the building (height \( h \)).
- Above this line, draw a smaller vertical line representing the chimney (height \( h_1 \)).
- The total height from the ground to the top of the chimney is \( h + h_1 \).
- Mark the ground point from where the angles of elevation are measured.

3. **Using Trigonometry**:
- For the angle of elevation to the top of the building:
\[ \tan(x) = \frac{h}{b} \quad \text{(where \( b \) is the horizontal distance from the point to the base of the building)} \]
- For the angle of elevation to the top of the chimney:
\[ \tan(45^\circ) = \frac{h + h_1}{b} \]
- Since \( \tan(45^\circ) = 1 \), we can simplify this to:
\[ h + h_1 = b \]

4. **Setting Up the Equations**:
- From the first equation:
\[ b = \frac{h}{\tan(x)} \]
- From the second equation:
\[ b = h + h_1 \]

5. **Equating the Two Expressions for \( b \)**:
- Set the two expressions for \( b \) equal to each other:
\[ \frac{h}{\tan(x)} = h + h_1 \]

6. **Solving for \( h_1 \)**:
- Rearranging the equation gives:
\[ h + h_1 = \frac{h}{\tan(x)} \]
- Isolate \( h_1 \):
\[ h_1 = \frac{h}{\tan(x)} - h \]
- Factor out \( h \):
\[ h_1 = h \left( \frac{1}{\tan(x)} - 1 \right) \]
- Rewrite using the cotangent function:
\[ h_1 = h \left( \cot(x) - 1 \right) \]

### Final Answer:
The height of the chimney \( h_1 \) in meters is:
\[ h_1 = h (\cot(x) - 1) \]