The angle of elevation of the Sun, when the length of the shadow of a tree is $\sqrt{3}$ times the height of the tree, is:

Question

Accepted Answer

To determine the angle of elevation of the Sun when the length of the shadow of a tree is $\sqrt{3}$ times the height of the tree, we can use trigonometry. Specifically, we will use the tangent function, which relates the angle of elevation to the opposite side (height of the tree) and the adjacent side (length of the shadow).

Let's denote:
- $ h $ as the height of the tree.
- $ s $ as the length of the shadow.
- $\theta$ as the angle of elevation of the Sun.

According to the problem, the length of the shadow $ s $ is $\sqrt{3}$ times the height of the tree $ h $. Therefore, we have:
$$ s = \sqrt{3} \times h $$

The tangent of the angle of elevation $\theta$ is given by the ratio of the opposite side to the adjacent side:
$$ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{s} $$

Substituting the given relationship $ s = \sqrt{3} \times h $ into the equation, we get:
$$ \tan(\theta) = \frac{h}{\sqrt{3} \times h} = \frac{1}{\sqrt{3}} $$

The angle $\theta$ for which $\tan(\theta) = \frac{1}{\sqrt{3}}$ is a well-known trigonometric value. It corresponds to:
$$ \theta = 30^{\circ} $$

Therefore, the correct answer is Option 1: $30^{\circ}$.

Now, let's explain why the other options are incorrect:

1. **Option 2: $45^{\circ}$** - If the angle of elevation were $45^{\circ}$, then the tangent of the angle would be 1, meaning the height of the tree and the length of the shadow would be equal. However, the problem states that the shadow is $\sqrt{3}$ times the height, not equal to it.

2. **Option 3: $60^{\circ}$** - If the angle of elevation were $60^{\circ}$, then the tangent of the angle would be $\sqrt{3}$, meaning the height of the tree would be $\sqrt{3}$ times the length of the shadow. This contradicts the given condition that the shadow is $\sqrt{3}$ times the height.

3. **Option 4: $75^{\circ}$** - If the angle of elevation were $75^{\circ}$, the tangent of the angle would be a value greater than $\sqrt{3}$, which does not match the condition that the shadow is $\sqrt{3}$ times the height of the tree.

In conclusion, the only angle that satisfies the condition given in the problem is $30^{\circ}$, making Option 1 the correct answer.