It is observed that on walking `x` meters towards a tower in a horizontal line through its base, the elevation of its top changes from `30^@` to `60^@`. The height of the tower is:

Options

A. `3sqrt2x`
B. `2sqrt3x`
C. `sqrt3/2x`
D. `2/sqrt3x`

Answer

To find the height of the tower based on the given angles of elevation, we can follow these steps:

### Step 1: Understand the problem
We have a tower and two positions from which the angles of elevation to the top of the tower are given: 30 degrees and 60 degrees. We denote the height of the tower as \( H \) and the horizontal distance from the base of the tower to the two positions as \( BC \) and \( BD \).

### Step 2: Set up the triangles
From the first position where the angle of elevation is 60 degrees, we can form a right triangle \( ABC \):
- The angle \( ACB = 60^\circ \)
- The height of the tower \( H \) is the opposite side to angle \( 60^\circ \)
- The distance \( BC \) is the adjacent side to angle \( 60^\circ \)

Using the tangent function:
\[ \tan(60^\circ) = \frac{H}{BC} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{H}{BC} \quad \Rightarrow \quad BC = \frac{H}{\sqrt{3}} \quad \text{(1)} \]

### Step 3: Set up the second triangle
From the second position where the angle of elevation is 30 degrees, we can form another right triangle \( ABD \):
- The angle \( ADB = 30^\circ \)
- The distance \( BD \) is the total distance from the base of the tower to the second position, which can be expressed as \( BC + x \).

Using the tangent function again:
\[ \tan(30^\circ) = \frac{H}{BD} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{H}{BD} \quad \Rightarrow \quad BD = \sqrt{3}H \quad \text{(2)} \]

### Step 4: Relate \( BD \) and \( BC \)
From the previous steps, we know:
\[ BD = BC + x \]
Substituting the expressions from (1) and (2):
\[ \sqrt{3}H = \frac{H}{\sqrt{3}} + x \]

### Step 5: Solve for \( x \)
Rearranging the equation:
\[ \sqrt{3}H - \frac{H}{\sqrt{3}} = x \]
To combine the terms on the left:
\[ x = \left(\sqrt{3} - \frac{1}{\sqrt{3}}\right)H = \left(\frac{3}{\sqrt{3}} - \frac{1}{\sqrt{3}}\right)H = \frac{2}{\sqrt{3}}H \]

### Step 6: Solve for \( H \)
Now, we can express \( H \) in terms of \( x \):
\[ H = \frac{\sqrt{3}}{2}x \]

### Conclusion
Thus, the height of the tower \( H \) is given by:
\[ H = \frac{\sqrt{3}}{2}x \]