The angle of elevation of the top of a tower at any point on the ground is `30^@` and moving 20 metres towards the tower it becomes `60^@`. The height of the tower is

Options

A. 10 m
B. `10sqrt3` m
C. `10/sqrt3` m
D. none of these

Answer

To solve the problem, we will use trigonometric ratios. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a tower (AB) of height \( H \). The angle of elevation from a point \( P \) on the ground to the top of the tower is \( 30^\circ \). When we move 20 meters towards the tower to a new point \( E \), the angle of elevation becomes \( 60^\circ \).

### Step 2: Set Up the Diagram
1. Let the distance from point \( P \) to the base of the tower \( A \) be \( x \) meters.
2. Therefore, the distance from point \( E \) to the base of the tower \( A \) will be \( x - 20 \) meters.

### Step 3: Use the Tangent Function for Point P
From point \( P \), we can write the equation using the tangent of the angle of elevation:
\[ \tan(30^\circ) = \frac{H}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{H}{x} \]
Cross-multiplying gives us:
\[ H = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

### Step 4: Use the Tangent Function for Point E
From point \( E \), we can write another equation:
\[ \tan(60^\circ) = \frac{H}{x - 20} \]
We know that \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{H}{x - 20} \]
Cross-multiplying gives us:
\[ H = \sqrt{3}(x - 20) \quad \text{(Equation 2)} \]

### Step 5: Set the Two Equations for H Equal
Now we have two expressions for \( H \):
1. From Equation 1: \( H = \frac{x}{\sqrt{3}} \)
2. From Equation 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 either equation for \( H \). Using Equation 1:
\[ H = \frac{30}{\sqrt{3}} = 10\sqrt{3} \]

### Conclusion
The height of the tower is \( H = 10\sqrt{3} \) meters.

---