A man standing 30ms South of a tower of height h walks 60 m to the East and finds the angle of elevation of the top of the tower to `30^(@)`. Find h.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
A man is standing 30 meters south of a tower of height \( h \). He walks 60 meters east and finds the angle of elevation to the top of the tower is \( 30^\circ \). We need to find the height \( h \) of the tower.

### Step 2: Draw a Diagram
1. Draw a vertical line representing the tower, labeled as \( OA \) where \( O \) is the base and \( A \) is the top of the tower.
2. Mark point \( B \) 30 meters south of point \( O \).
3. From point \( B \), draw a line 60 meters to the east to point \( C \).
4. The angle of elevation from point \( C \) to point \( A \) is \( 30^\circ \).

### Step 3: Identify the Triangle
In triangle \( ABC \):
- \( AB \) is the height of the tower \( h \).
- \( BC \) is the distance walked east, which is 60 m.
- \( OA \) is the height of the tower \( h \).
- \( OB \) is the distance south, which is 30 m.

### Step 4: Calculate the Length of \( AC \)
Using the Pythagorean theorem in triangle \( ABC \):
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ AC^2 = h^2 + 60^2 \]
\[ AC^2 = h^2 + 3600 \]

### Step 5: Find the Length of \( AC \) Using the Angle of Elevation
From point \( C \), the angle of elevation to the top of the tower is \( 30^\circ \). We can use the tangent function:
\[ \tan(30^\circ) = \frac{h}{AC} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{AC} \]
This implies:
\[ h = AC \cdot \frac{1}{\sqrt{3}} \]

### Step 6: Calculate \( AC \)
To find \( AC \), we can use the coordinates:
- The horizontal distance from \( O \) to \( C \) is \( 60 \) m (east) and the vertical distance from \( O \) to \( B \) is \( 30 \) m (south).
Using the Pythagorean theorem:
\[ AC = \sqrt{(60)^2 + (30)^2} = \sqrt{3600 + 900} = \sqrt{4500} = 30\sqrt{5} \]

### Step 7: Substitute \( AC \) back into the height equation
Now substitute \( AC \) into the height equation:
\[ h = 30\sqrt{5} \cdot \frac{1}{\sqrt{3}} = 30 \cdot \frac{\sqrt{5}}{\sqrt{3}} = 10 \cdot \sqrt{15} \]

### Final Answer
Thus, the height \( h \) of the tower is:
\[ h = 10\sqrt{15} \text{ meters} \]