The angle of depression of a car standing on the ground from the top of a 66 m tower, is `30^(@).` Find the distance of the car from the base of the tower.

Answer

To solve the problem, we will use the concept of trigonometry, specifically the tangent function, which relates the angle of a right triangle to the opposite and adjacent sides.

### Step-by-Step Solution:

1. **Understand the Scenario**:
We have a tower of height 66 meters. The angle of depression from the top of the tower to the car on the ground is 30 degrees.

2. **Draw a Diagram**:
Visualize the situation by drawing a right triangle. The tower represents the vertical side (height), the distance from the base of the tower to the car represents the horizontal side (adjacent side), and the line of sight from the top of the tower to the car represents the hypotenuse.

3. **Identify the Triangle**:
In the right triangle formed, let:
- AB = height of the tower = 66 m (vertical side)
- BC = distance from the base of the tower to the car (horizontal side, which we need to find)
- Angle A = angle of depression = 30 degrees

4. **Apply the Tangent Function**:
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. Here, we can write:
\[ \tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{66}{BC} \]

5. **Calculate \(\tan(30^\circ)\)**:
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \text{ or approximately } 0.577 \]
Therefore, we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{66}{BC} \]

6. **Rearranging the Equation**:
To find BC (the distance from the base of the tower to the car), we can rearrange the equation:
\[ BC = 66 \cdot \sqrt{3} \]

7. **Calculate the Distance**:
Now, we can calculate the value:
\[ BC \approx 66 \cdot 1.732 \approx 114.49 \text{ meters} \]

### Final Answer:
The distance of the car from the base of the tower is approximately **114.49 meters**.