A man observes the angle of elevation of the top of a building to be `30^(@)`. He walks towards it in a horizontal line through its base. On covering 60 m the angle of elevation changes to `60^(@)`. Find the height of the building correct to the nearest metre.

Answer

To solve the problem step by step, we will use trigonometric ratios to find the height of the building.

### Step 1: Define the problem
Let:
- \( h \) = height of the building
- \( x \) = initial horizontal distance from the man to the base of the building
- The man observes the angle of elevation of the top of the building to be \( 30^\circ \) from point A, and after walking 60 m towards the building, the angle of elevation changes to \( 60^\circ \) from point B.

### Step 2: Set up the equations using tangent
From point A (where the angle of elevation is \( 30^\circ \)):
\[ \tan(30^\circ) = \frac{h}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \quad \Rightarrow \quad h = \frac{x}{\sqrt{3}} \quad \text{(1)} \]

From point B (where the angle of elevation is \( 60^\circ \)):
\[ \tan(60^\circ) = \frac{h}{x - 60} \]
We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h}{x - 60} \quad \Rightarrow \quad h = \sqrt{3}(x - 60) \quad \text{(2)} \]

### Step 3: Set the equations equal to each other
From equations (1) and (2):
\[ \frac{x}{\sqrt{3}} = \sqrt{3}(x - 60) \]

### Step 4: Solve for \( x \)
Multiply both sides by \( \sqrt{3} \) to eliminate the fraction:
\[ x = 3(x - 60) \]
Distributing the right side:
\[ x = 3x - 180 \]
Rearranging gives:
\[ 180 = 3x - x \quad \Rightarrow \quad 180 = 2x \quad \Rightarrow \quad x = 90 \]

### Step 5: Find the height \( h \)
Substituting \( x = 90 \) back into equation (1):
\[ h = \frac{90}{\sqrt{3}} = 30\sqrt{3} \]

### Step 6: Calculate the numerical value of \( h \)
Using \( \sqrt{3} \approx 1.732 \):
\[ h \approx 30 \times 1.732 \approx 51.96 \]
Rounding to the nearest metre gives:
\[ h \approx 52 \text{ m} \]

### Final Answer:
The height of the building is approximately **52 metres**.