A hunter on top of a tree sees an antelope at an angle of depression of $30^{\circ}$. The height of the tree is 8 m. We need to find the distance between the hunter and the antelope.

1. Problem Description

A hunter on top of a tree sees an antelope at an angle of depression of $30^{\circ}$. The height of the tree is 8 m. We need to find the distance between the hunter and the antelope.

2. Solution Steps

Let $h$ be the height of the tree, which is 8 m.

Let $d$ be the horizontal distance between the base of the tree and the antelope.

Let $s$ be the straight-line distance between the hunter and the antelope.

The angle of depression is the angle between the horizontal line from the hunter's eye and the line of sight to the antelope. In this case, the angle of depression is $30^{\circ}$.

Since the angle of depression is $30^{\circ}$, the angle of elevation from the antelope to the hunter is also $30^{\circ}$. This forms a right triangle with the height of the tree as the opposite side and the horizontal distance $d$ as the adjacent side to the $30^{\circ}$ angle.

We can use the tangent function to relate the angle of elevation, the height of the tree, and the horizontal distance.

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

$\tan(30^{\circ}) = \frac{h}{d} = \frac{8}{d}$

$d = \frac{8}{\tan(30^{\circ})}$

Since $\tan(30^{\circ}) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$, we have

$d = \frac{8}{\frac{1}{\sqrt{3}}} = 8\sqrt{3}$

Now, we want to find the distance $s$ between the hunter and the antelope. We can use the cosine function to relate the angle of depression, the distance $s$, and the height of the tree.

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

$\sin(30^{\circ}) = \frac{8}{s}$

$s = \frac{8}{\sin(30^{\circ})}$

Since $\sin(30^{\circ}) = \frac{1}{2}$, we have

$s = \frac{8}{\frac{1}{2}} = 8 \cdot 2 = 16$

Alternatively, we can use the Pythagorean theorem: $s^2 = h^2 + d^2$.

$s^2 = 8^2 + (8\sqrt{3})^2 = 64 + 64 \cdot 3 = 64 + 192 = 256$

$s = \sqrt{256} = 16$

3. Final Answer

The distance between the hunter and the antelope is 16 m.