The angle of elevation of the top of a tower from a point on the ground is `30^(@)`. After walking 45 m towards the tower, the angle of elevation becomes `45^(@)`. Find the height of the tower.

Answer

To solve the problem, we will use trigonometric ratios and the properties of right triangles. Let's denote the height of the tower as \( h \).

### Step-by-Step Solution:

1. **Identify the Points and Angles:**
- Let point \( A \) be the top of the tower, point \( B \) be the base of the tower, and point \( C \) be the point on the ground where the observer initially stands.
- The angle of elevation from point \( C \) to point \( A \) is \( 30^\circ \).
- After walking 45 meters towards the tower to point \( D \), the angle of elevation becomes \( 45^\circ \).

2. **Set Up the Triangles:**
- In triangle \( ABC \) (where \( C \) is the observer's initial position):
\[ \tan(30^\circ) = \frac{h}{BC} \]
- In triangle \( ABD \) (where \( D \) is the observer's new position):
\[ \tan(45^\circ) = \frac{h}{BD} \]

3. **Calculate the Tangent Values:**
- We know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \quad \text{and} \quad \tan(45^\circ) = 1 \]

4. **Express \( BC \) and \( BD \):**
- From triangle \( ABD \):
\[ \tan(45^\circ) = 1 \Rightarrow 1 = \frac{h}{BD} \Rightarrow BD = h \]
- From triangle \( ABC \):
\[ \tan(30^\circ) = \frac{h}{BC} \Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{BC} \Rightarrow BC = h \sqrt{3} \]

5. **Relate \( BD \) and \( BC \):**
- Since \( BD = BC - 45 \):
\[ h = h \sqrt{3} - 45 \]

6. **Rearranging the Equation:**
- Rearranging gives:
\[ h \sqrt{3} - h = 45 \Rightarrow h(\sqrt{3} - 1) = 45 \]

7. **Solve for \( h \):**
- Thus, we have:
\[ h = \frac{45}{\sqrt{3} - 1} \]
- To simplify, multiply the numerator and denominator by the conjugate:
\[ h = \frac{45(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{45(\sqrt{3} + 1)}{3 - 1} = \frac{45(\sqrt{3} + 1)}{2} \]
- Substituting \( \sqrt{3} \approx 1.732 \):
\[ h = \frac{45(1.732 + 1)}{2} = \frac{45(2.732)}{2} = 61.47 \text{ meters} \]

### Final Answer:
The height of the tower is approximately \( 61.47 \) meters.