From the top of a house A in a street, the angles of elevation and depression of the top and foot of another house B on the opposite side of the street are `60^(@)` and `45^(@)` , respectively. If the height of house A is 36 m, then what is the height of house B? (Your answer should be nearest to an Integer.

Options

A. 91 cm
B. 93 m
C. 94 m
D. 98 m

Answer

To solve the problem, we will use trigonometry, specifically the tangent function, which relates the angles of elevation and depression to the heights and distances involved.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have two houses, A and B. The height of house A (AC) is given as 36 m.
- The angle of elevation to the top of house B (point D) from the top of house A (point C) is 60°.
- The angle of depression to the foot of house B (point E) from the top of house A (point C) is 45°.

2. **Draw a Diagram**:
- Draw a horizontal line representing the ground.
- Mark point C at the top of house A (height = 36 m).
- Mark point A at the base of house A, point B at the base of house B, point D at the top of house B, and point E at the foot of house B.

3. **Identify the Triangles**:
- Triangle CBE for the angle of depression (45°).
- Triangle CDE for the angle of elevation (60°).

4. **Calculate the Distance from A to B**:
- For triangle CBE (angle of depression = 45°):
- Since tan(45°) = 1, we have:
\[ \text{Height of A to E} = \text{Distance from A to B} \]
Let this distance be \(d\).
Therefore, \(d = 36\) m (the height of house A).

5. **Calculate the Height of House B (BD)**:
- For triangle CDE (angle of elevation = 60°):
- We know that:
\[ \tan(60°) = \frac{\text{Height of D (BD)}}{\text{Distance from A to B (d)}} \]
Since \(\tan(60°) = \sqrt{3}\), we can write:
\[ \sqrt{3} = \frac{BD}{d} \]
Substituting \(d = 36\):
\[ BD = 36 \cdot \sqrt{3} \]

6. **Calculate the Total Height of House B**:
- The total height of house B (BD) is the sum of the height of house A (36 m) and the height from E to D (BD):
\[ \text{Total Height of House B} = BD + AE \]
where \(AE = 36\) m (the height of house A).
- Therefore:
\[ \text{Total Height of House B} = 36 + 36\sqrt{3} \]

7. **Substituting the Value of \(\sqrt{3}\)**:
- Using \(\sqrt{3} \approx 1.732\):
\[ BD = 36 \cdot 1.732 \approx 62.592 \]
- Thus, the total height of house B becomes:
\[ \text{Total Height of House B} = 36 + 62.592 \approx 98.592 \text{ m} \]

8. **Rounding to the Nearest Integer**:
- The height of house B rounded to the nearest integer is **99 m**.

### Final Answer:
The height of house B is approximately **99 m**.