The angle of elevation of top of a tower from a point on the ground is `30^@` and it is `60^@` when it is viewed from a point located 40 m away from the initial point towards the tower the height of the tower is

Options

A. `20sqrt3` m
B. `sqrt3/20` m
C. `10sqrt3`m
D. `sqrt3/10` m

Explanation

To solve the problem, we need to find the height of the tower using the given angles of elevation and the distance between the two points from which the angles are measured.

### Step 1: Define the problem
Let the height of the tower be \( h \).
Let the distance from the first point \( P \) to the base of the tower \( B \) be \( x \).
The distance from the second point \( E \) to the base of the tower \( B \) is \( x - 40 \) since point \( E \) is 40 m closer to the tower than point \( P \).

### Step 2: Use the first angle of elevation
From point \( P \), the angle of elevation to the top of the tower is \( 30^\circ \).
Using the tangent function:
\[ \tan(30^\circ) = \frac{h}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{h}{x} = \frac{1}{\sqrt{3}} \]
This gives us:
\[ h = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

### Step 3: Use the second angle of elevation
From point \( E \), the angle of elevation to the top of the tower is \( 60^\circ \).
Using the tangent function again:
\[ \tan(60^\circ) = \frac{h}{x - 40} \]
We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \frac{h}{x - 40} = \sqrt{3} \]
This gives us:
\[ h = \sqrt{3}(x - 40) \quad \text{(Equation 2)} \]

### Step 4: Set the equations equal to each other
Now we have two expressions for \( h \):
From Equation 1:
\[ h = \frac{x}{\sqrt{3}} \]
From Equation 2:
\[ h = \sqrt{3}(x - 40) \]
Setting them equal:
\[ \frac{x}{\sqrt{3}} = \sqrt{3}(x - 40) \]

### Step 5: Solve for \( x \)
Multiply both sides by \( \sqrt{3} \) to eliminate the fraction:
\[ x = 3(x - 40) \]
Expanding the right side:
\[ x = 3x - 120 \]
Rearranging gives:
\[ 120 = 3x - x \]
\[ 120 = 2x \]
\[ x = 60 \]

### Step 6: Find the height \( h \)
Substituting \( x = 60 \) back into Equation 1:
\[ h = \frac{60}{\sqrt{3}} = 20\sqrt{3} \]

### Conclusion
The height of the tower is \( 20\sqrt{3} \) meters.