The angle of elevation of the top of a tower at a point A on the ground is `30^(@)`. On walking 20 meters toward the tower, the angle of elevation is `60^(@)`. Find the height of the tower and its distance from A.

Answer

To solve the problem step by step, we will use trigonometric relationships involving the angles of elevation.

### Step 1: Understand the Problem
We have a tower and two points on the ground where the angles of elevation to the top of the tower are given. The first angle of elevation from point A is \(30^\circ\), and after walking 20 meters towards the tower, the angle of elevation becomes \(60^\circ\). We need to find the height of the tower and the distance from point A to the base of the tower.

### Step 2: Set Up the Diagram
Let's denote:
- The height of the tower as \(h\).
- The distance from point A to the base of the tower as \(x\).
- After walking 20 meters towards the tower, the new distance from point B (the new position) to the base of the tower is \(x - 20\).

### Step 3: Use the First Angle of Elevation
From point A, using the angle of elevation \(30^\circ\):
\[ \tan(30^\circ) = \frac{h}{x} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so we can write:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \]
This gives us:
\[ h = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

### Step 4: Use the Second Angle of Elevation
From point B, using the angle of elevation \(60^\circ\):
\[ \tan(60^\circ) = \frac{h}{x - 20} \]
We know that \(\tan(60^\circ) = \sqrt{3}\), so we can write:
\[ \sqrt{3} = \frac{h}{x - 20} \]
This gives us:
\[ h = \sqrt{3}(x - 20) \quad \text{(Equation 2)} \]

### Step 5: Set the Two Equations for Height Equal
Now we have two expressions for \(h\):
1. \(h = \frac{x}{\sqrt{3}}\)
2. \(h = \sqrt{3}(x - 20)\)

Setting them equal to each other:
\[ \frac{x}{\sqrt{3}} = \sqrt{3}(x - 20) \]

### Step 6: Solve for \(x\)
Multiply both sides by \(\sqrt{3}\) to eliminate the fraction:
\[ x = 3(x - 20) \]
Expanding the right side:
\[ x = 3x - 60 \]
Rearranging gives:
\[ 60 = 3x - x \]
\[ 60 = 2x \]
\[ x = 30 \]

### Step 7: Find the Height \(h\)
Now substitute \(x = 30\) back into Equation 1 to find \(h\):
\[ h = \frac{30}{\sqrt{3}} = 10\sqrt{3} \]

### Step 8: Conclusion
The height of the tower is \(10\sqrt{3}\) meters, and the distance from point A to the base of the tower is \(30\) meters.

### Summary of Results
- Height of the tower \(h = 10\sqrt{3}\) meters
- Distance from point A to the base of the tower \(x = 30\) meters