A man observes the angle of elevation of the top of the tower to be `45^(@)`. He walks towards it in a horizontal line through its base. On covering 20 m the angle of elevation change to `60^(@)`. Find the height of the tower correct to 2 significant figures.

Answer

To solve the problem, we will use trigonometric ratios. Let's break down the solution step by step.

### Step 1: Draw the Diagram
First, we need to visualize the problem. Draw a vertical line representing the tower. Let the height of the tower be \( h \).

- At point A (the initial position of the man), the angle of elevation to the top of the tower is \( 45^\circ \).
- The man walks towards the tower and reaches point B, where the angle of elevation is \( 60^\circ \).
- The horizontal distance from point A to the base of the tower is \( x \).
- The distance the man walks towards the tower is \( 20 \, \text{m} \), so the distance from point B to the base of the tower is \( x - 20 \).

### Step 2: Set Up the Equations
Using the tangent function, we can set up two equations based on the angles of elevation.

1. From point A:
\[ \tan(45^\circ) = \frac{h}{x} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ h = x \]

2. From point B:
\[ \tan(60^\circ) = \frac{h}{x - 20} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ h = \sqrt{3}(x - 20) \]

### Step 3: Substitute and Solve
Now we can substitute \( h \) from the first equation into the second equation:
\[ x = \sqrt{3}(x - 20) \]

Expanding this gives:
\[ x = \sqrt{3}x - 20\sqrt{3} \]

Now, rearranging the equation:
\[ x - \sqrt{3}x = -20\sqrt{3} \]
\[ x(1 - \sqrt{3}) = -20\sqrt{3} \]
\[ x = \frac{-20\sqrt{3}}{1 - \sqrt{3}} \]

### Step 4: Rationalize the Denominator
To simplify \( x \), we rationalize the denominator:
\[ x = \frac{-20\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \]
\[ = \frac{-20\sqrt{3}(1 + \sqrt{3})}{1 - 3} \]
\[ = \frac{-20\sqrt{3}(1 + \sqrt{3})}{-2} \]
\[ = 10\sqrt{3}(1 + \sqrt{3}) \]

### Step 5: Substitute Back to Find \( h \)
Now substitute \( x \) back into \( h = x \):
\[ h = 10\sqrt{3}(1 + \sqrt{3}) \]
Calculating \( \sqrt{3} \approx 1.732 \):
\[ h = 10 \times 1.732 \times (1 + 1.732) \]
\[ = 10 \times 1.732 \times 2.732 \]
Calculating this gives:
\[ h \approx 10 \times 1.732 \times 2.732 \approx 47.32 \, \text{m} \]

### Final Answer
The height of the tower is approximately \( 47.32 \, \text{m} \), which can be rounded to \( 47 \, \text{m} \) when expressed to two significant figures.