The angle of elevation of the top of a tower at a point on the level ground is 30°. After walking a distance of I 00 m towards the foot of the tower along the horizontal line through the foot of the tower on the same level ground, the angle of elevation of the top of the tower is 60°. Find the height of the tower.

Answer

To find the height of the tower, we can use trigonometric ratios based on the angles of elevation given in the problem. Let's break down the solution step by step.

### Step 1: Define the problem
Let:
- \( H \) = height of the tower
- \( A \) = point where the angle of elevation is 30°
- \( B \) = foot of the tower
- \( C \) = top of the tower
- \( D \) = point after walking 100 m towards the tower

### Step 2: Set up the first triangle (30° angle)
From point \( A \), the angle of elevation to the top of the tower \( C \) is 30°.
Using the tangent function:
\[ \tan(30°) = \frac{H}{AB} \]
Where \( AB \) is the distance from point \( A \) to the foot of the tower \( B \).

Since \( \tan(30°) = \frac{1}{\sqrt{3}} \), we can write:
\[ \frac{1}{\sqrt{3}} = \frac{H}{AB} \]
Thus, we have:
\[ AB = H \cdot \sqrt{3} \]

### Step 3: Set up the second triangle (60° angle)
From point \( D \), after walking 100 m towards the tower, the angle of elevation to the top of the tower \( C \) is 60°.
Using the tangent function again:
\[ \tan(60°) = \frac{H}{DB} \]
Where \( DB \) is the distance from point \( D \) to the foot of the tower \( B \).

Since \( DB = AB - 100 \), we can express this as:
\[ \tan(60°) = \frac{H}{AB - 100} \]
Given that \( \tan(60°) = \sqrt{3} \), we write:
\[ \sqrt{3} = \frac{H}{AB - 100} \]
Thus:
\[ AB - 100 = \frac{H}{\sqrt{3}} \]

### Step 4: Substitute \( AB \) in the second equation
Now we have two equations:
1. \( AB = H \cdot \sqrt{3} \)
2. \( AB - 100 = \frac{H}{\sqrt{3}} \)

Substituting the first equation into the second:
\[ H \cdot \sqrt{3} - 100 = \frac{H}{\sqrt{3}} \]

### Step 5: Solve for \( H \)
Multiply through by \( \sqrt{3} \) to eliminate the fraction:
\[ 3H - 100\sqrt{3} = H \]
Rearranging gives:
\[ 3H - H = 100\sqrt{3} \]
\[ 2H = 100\sqrt{3} \]
\[ H = 50\sqrt{3} \]

### Conclusion
The height of the tower is:
\[ H = 50\sqrt{3} \text{ meters} \]