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 :

Question

Accepted Answer

To determine the height of the tower given the change in elevation from 30° to 60° as one walks x meters towards it, we can use trigonometric principles.

### Explanation for Option 3:

1. **Initial Position (Elevation 30°):**
   - Let the distance from the initial position to the base of the tower be $ d $.
   - The height of the tower $ h $ can be expressed using the tangent of the angle of elevation:
     $$
     \tan(30^\circ) = \frac{h}{d}
     $$
   - Since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, we have:
     $$
     \frac{1}{\sqrt{3}} = \frac{h}{d} \implies h = \frac{d}{\sqrt{3}}
     $$

2. **Position After Walking x Meters (Elevation 60°):**
   - The new distance from the tower is $ d - x $.
   - Using the tangent of the new angle of elevation:
     $$
     \tan(60^\circ) = \frac{h}{d-x}
     $$
   - Since $\tan(60^\circ) = \sqrt{3}$, we have:
     $$
     \sqrt{3} = \frac{h}{d-x} \implies h = \sqrt{3}(d-x)
     $$

3. **Equating the Two Expressions for h:**
   - From the two equations for $ h $, we have:
     $$
     \frac{d}{\sqrt{3}} = \sqrt{3}(d-x)
     $$
   - Solving for $ d $:
     $$
     d = 3(d-x)
     $$
     $$
     d = 3d - 3x
     $$
     $$
     2d = 3x
     $$
     $$
     d = \frac{3x}{2}
     $$

4. **Substitute $ d $ Back to Find $ h $:**
   - Using $ h = \frac{d}{\sqrt{3}} $:
     $$
     h = \frac{\frac{3x}{2}}{\sqrt{3}} = \frac{3x}{2\sqrt{3}} = \frac{\sqrt{3}}{2}x
     $$

Thus, the height of the tower is $\frac{\sqrt{3}}{2}x$, which corresponds to **Option 3**.

### Explanation for Why Other Options are Incorrect:

- **Option 1: $3\sqrt{2}x$**
  - This option does not align with the derived formula using trigonometric identities and the given angles. The calculations show that the height is not proportional to $3\sqrt{2}x$.

- **Option 2: $2\sqrt{3}x$**
  - This option suggests a different proportionality that does not match the derived relationship between the height and the distance walked. The calculations clearly show a different factor.

- **Option 4: $\frac{2}{\sqrt{3}}x$**
  - This option represents an incorrect simplification of the trigonometric relationships involved. The correct simplification leads to $\frac{\sqrt{3}}{2}x$, not $\frac{2}{\sqrt{3}}x$.

In conclusion, the correct answer is indeed **Option 3: $\frac{\sqrt{3}}{2}x$**, as it accurately reflects the height of the tower based on the given conditions and trigonometric calculations.