Find the angle of elevation of the Sun, when the length of the shadow of a tree is √ 3 times the height of the tree.

The correct option is 30^◦.

Let us understand the problem by modeling it geometrically.

Let the height of the tree be represented by h, and let the length of the shadow cast by the tree on the ground be represented by s.

According to the given problem, the length of the shadow is $\sqrt{3}$ times the height of the tree. Therefore, we can write the relationship as:

$s=\sqrt{3}h$

If we represent the tree and its shadow as a right-angled triangle, the tree is the perpendicular side (opposite side) of height h, and the shadow is the base side (adjacent side) of length s. Let the angle of elevation of the Sun be represented by $\theta$.

Using basic trigonometry, the tangent of the angle of elevation $\theta$ is defined as the ratio of the height of the tree (opposite side) to the length of the shadow (adjacent side):

$tan\left(\theta \right)=\frac{h}{s}$

Now, we substitute the value of s in terms of h into the trigonometric ratio:

$tan\left(\theta \right)=\frac{h}{\sqrt{3}h}$

By simplifying the fraction, the height variable h cancels out, giving:

$tan\left(\theta \right)=\frac{1}{\sqrt{3}}$

We know from standard trigonometric values that the tangent of 30^◦ is equal to $\frac{1}{\sqrt{3}}$. Therefore, we have:

$\theta ={30}^{◦}$

Thus, the angle of elevation of the Sun is 30^◦.