If the length of the shadow of a vertical tower is $\frac{1}{\sqrt{3}}$ times its height, then the angle of elevation of the sun is :

Question

Accepted Answer

To determine the angle of elevation of the sun when the length of the shadow of a vertical tower is $\frac{1}{\sqrt{3}}$ times its height, we can use trigonometry, specifically the tangent function.

### Explanation:

1. **Understanding the Problem:**
   - Let the height of the tower be $ h $.
   - The length of the shadow is $\frac{1}{\sqrt{3}} \times h$.
   - The angle of elevation of the sun is the angle $\theta$ between the line of sight from the top of the tower to the tip of the shadow and the horizontal ground.

2. **Using Trigonometry:**
   - The tangent of the angle of elevation $\theta$ is given by the ratio of the opposite side (height of the tower) to the adjacent side (length of the shadow).
   - Therefore, $\tan(\theta) = \frac{\text{height of the tower}}{\text{length of the shadow}} = \frac{h}{\frac{1}{\sqrt{3}} \times h} = \sqrt{3}$.

3. **Determining the Angle:**
   - We know that $\tan(60^\circ) = \sqrt{3}$.
   - Therefore, the angle of elevation $\theta$ is $60^\circ$.

### Justification for Option 3:

- **Option 3 (60°) is correct** because the calculation shows that when the tangent of the angle is $\sqrt{3}$, the angle is indeed $60^\circ$.

### Explanation for Incorrect Options:

- **Option 1 (30°):** 
  - $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, which does not match the given condition where $\tan(\theta) = \sqrt{3}$.

- **Option 2 (45°):**
  - $\tan(45^\circ) = 1$, which does not satisfy the condition $\tan(\theta) = \sqrt{3}$.

- **Option 4 (90°):**
  - $\tan(90^\circ)$ is undefined, as the tangent function approaches infinity. This does not match the given condition.

In conclusion, the correct angle of elevation of the sun, given the problem's conditions, is $60^\circ$, which corresponds to Option 3.