The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is $30^{\circ}$ than when it is $60^{\circ}.$ The height of the tower is:

Question

Accepted Answer

To determine the height of the tower, we can use trigonometric principles related to the angles of elevation and the lengths of shadows.

Let's denote:
- $ h $ as the height of the tower.
- $ x $ as the length of the shadow when the Sun's altitude is $ 60^\circ $.
- $ x + 40 $ as the length of the shadow when the Sun's altitude is $ 30^\circ $.

Using the tangent function, which relates the angle of elevation to the opposite side (height of the tower) and the adjacent side (length of the shadow), we have:

1. When the Sun's altitude is $ 60^\circ $:
   $$
   \tan 60^\circ = \frac{h}{x}
   $$
   Since $\tan 60^\circ = \sqrt{3}$, we have:
   $$
   \sqrt{3} = \frac{h}{x} \quad \Rightarrow \quad h = x\sqrt{3}
   $$

2. When the Sun's altitude is $ 30^\circ $:
   $$
   \tan 30^\circ = \frac{h}{x + 40}
   $$
   Since $\tan 30^\circ = \frac{1}{\sqrt{3}}$, we have:
   $$
   \frac{1}{\sqrt{3}} = \frac{h}{x + 40} \quad \Rightarrow \quad h = \frac{x + 40}{\sqrt{3}}
   $$

Equating the two expressions for $ h $:
$$
x\sqrt{3} = \frac{x + 40}{\sqrt{3}}
$$

Multiplying through by $\sqrt{3}$ to eliminate the fraction:
$$
3x = x + 40
$$

Solving for $ x $:
$$
3x - x = 40 \quad \Rightarrow \quad 2x = 40 \quad \Rightarrow \quad x = 20
$$

Substitute $ x = 20 $ back into the expression for $ h $:
$$
h = x\sqrt{3} = 20\sqrt{3}
$$

Thus, the height of the tower is $ 20\sqrt{3} $ meters, which corresponds to Option 3.

**Explanation for Incorrect Options:**

1. **Option 1: $\sqrt{3}$ m**
   - This option is incorrect because it does not satisfy the trigonometric relationships derived from the given angles of elevation. The calculations show that the height is significantly larger than $\sqrt{3}$ meters.

2. **Option 2: $10\sqrt{3}$ m**
   - This option is incorrect because it represents half of the correct height. The calculations clearly show that the height is $20\sqrt{3}$ meters, not $10\sqrt{3}$.

3. **Option 4: 20 m**
   - This option is incorrect because it does not account for the $\sqrt{3}$ factor that arises from the trigonometric calculations. The height must be expressed in terms of $\sqrt{3}$ due to the angles involved.

Therefore, the correct answer is Option 3: $20\sqrt{3}$ meters.