The angle of elevation of the top of a tower from the top and bottom of a building of height 'a' are `30^@` and `45^@` respectively. If the tower and the building stand at the same level , the height of the tower is

Options

A. `a(3+sqrt(3))/2`
B. `a(sqrt(3)+1)`
C. `asqrt(3)`
D. `a/(sqrt(3)-1)`

Answer

To solve the problem, we will use trigonometric ratios and the properties of right triangles. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a building of height \( a \).
- The angle of elevation to the top of the tower from the top of the building is \( 30^\circ \).
- The angle of elevation to the top of the tower from the bottom of the building is \( 45^\circ \).
- We need to find the height of the tower, denoted as \( h \).

2. **Drawing the Diagram**:
- Let \( B \) be the bottom of the building, \( A \) the top of the building, and \( T \) the top of the tower.
- The height of the building \( AB = a \).
- The height of the tower \( BT = h \).
- The distance from the base of the building to the base of the tower is \( d \).

3. **Using the Angle of Elevation from the Bottom of the Building**:
- From point \( B \) (the bottom of the building), the angle of elevation to point \( T \) (the top of the tower) is \( 45^\circ \).
- Using the tangent function:
\[ \tan(45^\circ) = \frac{h}{d} \]
- Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h}{d} \implies h = d \quad \text{(1)} \]

4. **Using the Angle of Elevation from the Top of the Building**:
- From point \( A \) (the top of the building), the angle of elevation to point \( T \) is \( 30^\circ \).
- The height from point \( A \) to point \( T \) is \( h - a \).
- Using the tangent function again:
\[ \tan(30^\circ) = \frac{h - a}{d} \]
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h - a}{d} \implies h - a = \frac{d}{\sqrt{3}} \quad \text{(2)} \]

5. **Substituting Equation (1) into Equation (2)**:
- From equation (1), we know \( d = h \).
- Substitute \( d \) in equation (2):
\[ h - a = \frac{h}{\sqrt{3}} \]

6. **Solving for \( h \)**:
- Rearranging gives:
\[ h - \frac{h}{\sqrt{3}} = a \]
- Factor out \( h \):
\[ h \left(1 - \frac{1}{\sqrt{3}}\right) = a \]
- Simplifying:
\[ h \left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right) = a \]
- Therefore:
\[ h = \frac{a \sqrt{3}}{\sqrt{3} - 1} \]

7. **Rationalizing the Denominator**:
- To rationalize:
\[ h = \frac{a \sqrt{3} (\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{a \sqrt{3} (\sqrt{3} + 1)}{3 - 1} = \frac{a \sqrt{3} (\sqrt{3} + 1)}{2} \]

### Final Answer:
The height of the tower \( h \) is:
\[ h = \frac{a \sqrt{3} (\sqrt{3} + 1)}{2} \]