A man 1.5 meter tall is `30sqrt3` meter away from a building. When he sees the top of building the angle of elevation is 30°. Find the height of the building?

Options

A. 48 m
B. 53.4m
C. 30 m
D. 31.5m

Answer

To find the height of the building given the information about the man and the angle of elevation, we can follow these steps:

### Step 1: Understand the scenario
We have a man who is 1.5 meters tall standing 30√3 meters away from a building. The angle of elevation from the man's eyes to the top of the building is 30°.

### Step 2: Set up the problem
Let:
- A = the position of the man
- B = the top of the building
- C = the point on the ground directly below B
- D = the height of the man (1.5 meters)
- EC = the height from the man's eye level to the top of the building (BC)
- BE = the height of the building

### Step 3: Use the tangent function
The angle of elevation (30°) allows us to use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
For our scenario:
\[ \tan(30°) = \frac{EC}{AC} \]
Where:
- EC is the height from the man's eye level to the top of the building
- AC is the distance from the man to the building (30√3 meters)

### Step 4: Calculate EC
We know that:
\[ \tan(30°) = \frac{1}{\sqrt{3}} \]
Thus, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{EC}{30\sqrt{3}} \]
Cross-multiplying gives:
\[ EC = 30\sqrt{3} \cdot \frac{1}{\sqrt{3}} = 30 \]

### Step 5: Calculate the total height of the building
The total height of the building (BE) is the sum of the height from the man's eye level to the top of the building (EC) and the height of the man (D):
\[ BE = EC + D = 30 + 1.5 = 31.5 \text{ meters} \]

### Step 6: Conclusion
The height of the building is 31.5 meters.

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