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 ratios to find the height of the tower and the distance from point A.

### Step 1: Define the variables
Let:
- \( H \) = height of the tower
- \( x \) = distance from point A to the base of the tower

### Step 2: Set up the first triangle (triangle ACD)
From point A, the angle of elevation to the top of the tower is \( 30^\circ \). According to the tangent function:
\[ \tan(30^\circ) = \frac{H}{x + 20} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \). Therefore, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{H}{x + 20} \]
Cross-multiplying gives us:
\[ H = \frac{1}{\sqrt{3}}(x + 20) \]
Multiplying both sides by \( \sqrt{3} \):
\[ \sqrt{3}H = x + 20 \quad \text{(Equation 1)} \]

### Step 3: Set up the second triangle (triangle BCD)
After walking 20 meters towards the tower, the angle of elevation from point B is \( 60^\circ \). Again using the tangent function:
\[ \tan(60^\circ) = \frac{H}{x} \]
We know that \( \tan(60^\circ) = \sqrt{3} \). Therefore:
\[ \sqrt{3} = \frac{H}{x} \]
Cross-multiplying gives us:
\[ H = \sqrt{3}x \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
Now we substitute \( H \) from Equation 2 into Equation 1:
\[ \sqrt{3}(\sqrt{3}x) = x + 20 \]
This simplifies to:
\[ 3x = x + 20 \]
Subtracting \( x \) from both sides:
\[ 2x = 20 \]
Dividing by 2:
\[ x = 10 \text{ meters} \]

### Step 5: Find the height of the tower
Now we can find \( H \) using Equation 2:
\[ H = \sqrt{3}x = \sqrt{3}(10) = 10\sqrt{3} \text{ meters} \]

### Step 6: Find the total distance from point A to the tower
The total distance from point A to the base of the tower is:
\[ x + 20 = 10 + 20 = 30 \text{ meters} \]

### Final Answers
- Height of the tower \( H = 10\sqrt{3} \) meters
- Distance from point A to the tower = 30 meters