The angle of elevation of the top of a building from the top and bottom of a tree are x and y repectively . If the height of the tree is h metre , then (in metre) the height of the building is

Options

A. `(h cotx)/(cotx +coty)`
B. `(h cot y)/(cot c +coty)`
C. `(h cot x)/(cotx-coty)`
D. `(h cot y)/(cotx-coty)`

Answer

To find the height of the building based on the given angles of elevation from the top and bottom of a tree, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the height of the tree be \( h \) meters.
- Let the distance from the base of the tree to the base of the building be \( b \) meters.
- Let the height of the building be \( a \) meters.
- The angle of elevation from the top of the tree to the top of the building is \( x \) degrees.
- The angle of elevation from the bottom of the tree to the top of the building is \( y \) degrees.

2. **Setting Up the Relationships**:
- From the top of the tree, the height of the building can be expressed using the tangent of angle \( x \):
\[ \tan(x) = \frac{a - h}{b} \quad \text{(1)} \]
- From the bottom of the tree, the height of the building can be expressed using the tangent of angle \( y \):
\[ \tan(y) = \frac{a}{b} \quad \text{(2)} \]

3. **Rearranging the Equations**:
- From equation (1):
\[ a - h = b \tan(x) \implies a = b \tan(x) + h \quad \text{(3)} \]
- From equation (2):
\[ a = b \tan(y) \quad \text{(4)} \]

4. **Equating the Two Expressions for \( a \)**:
- Set equations (3) and (4) equal to each other:
\[ b \tan(x) + h = b \tan(y) \]

5. **Solving for \( b \)**:
- Rearranging gives:
\[ h = b \tan(y) - b \tan(x) \implies h = b (\tan(y) - \tan(x)) \]
- Thus, we can express \( b \) in terms of \( h \):
\[ b = \frac{h}{\tan(y) - \tan(x)} \quad \text{(5)} \]

6. **Substituting \( b \) Back to Find \( a \)**:
- Substitute \( b \) from equation (5) into equation (4):
\[ a = \frac{h}{\tan(y) - \tan(x)} \tan(y) \]
- This simplifies to:
\[ a = \frac{h \tan(y)}{\tan(y) - \tan(x)} \]

### Final Result:
The height of the building \( a \) in meters is given by:
\[ a = \frac{h \tan(y)}{\tan(y) - \tan(x)} \]