From a tower 18 m high the angle of elevation of the top of a tall building is `45^(@)` and the angle of depression of the bottom of the same building is `60^(@)`. What is the height of the building in metres?

Options

A. `12 + 6 sqrt(3)`
B. `6(3+ sqrt(3))`
C. `18 + sqrt(2)`
D. `6 (3+ (sqrt(3))/(2))`

Answer

To solve the problem of finding the height of the building, we will use trigonometric principles based on the angles of elevation and depression given in the question.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a tower of height 18 m.
- The angle of elevation to the top of the building is 45°.
- The angle of depression to the bottom of the building is 60°.
- We need to find the height of the building.

2. **Drawing the Diagram**:
- Let point A be the top of the tower, point B be the bottom of the tower, point C be the top of the building, and point D be the bottom of the building.
- The height of the tower (AB) = 18 m.
- The angle of elevation from A to C (top of the building) = 45°.
- The angle of depression from A to D (bottom of the building) = 60°.

3. **Identifying the Triangles**:
- From point A, we can form two right triangles:
- Triangle ACD for the angle of elevation (45°).
- Triangle ABD for the angle of depression (60°).

4. **Using the Angle of Elevation (45°)**:
- In triangle ACD, using the tangent function:
\[ \tan(45°) = \frac{h}{x} \]
where \( h \) is the height of the building above the tower (CD) and \( x \) is the horizontal distance from the tower to the building.
- Since \(\tan(45°) = 1\):
\[ 1 = \frac{h}{x} \implies h = x \]

5. **Using the Angle of Depression (60°)**:
- In triangle ABD, using the tangent function:
\[ \tan(60°) = \frac{18}{x} \]
- Since \(\tan(60°) = \sqrt{3}\):
\[ \sqrt{3} = \frac{18}{x} \implies x = \frac{18}{\sqrt{3}} = 6\sqrt{3} \]

6. **Finding the Height of the Building**:
- From the previous step, we know \( h = x \):
\[ h = 6\sqrt{3} \]
- The total height of the building (CD) is:
\[ CD = AB + h = 18 + 6\sqrt{3} \]

7. **Final Calculation**:
- Therefore, the height of the building is:
\[ \text{Height of the building} = 18 + 6\sqrt{3} \text{ meters} \]

### Conclusion:
The height of the building is \( 18 + 6\sqrt{3} \) meters.