From the top of a tower 180 m high, it was observed that the angle of depression of the bottom of a cat sitting on the ground was `30^(@)`. Find the distance of the cat from the foot of the tower.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have a tower that is 180 meters high, and from the top of this tower, the angle of depression to the cat sitting on the ground is 30 degrees. We need to find the horizontal distance from the foot of the tower to the cat.

### Step 2: Draw a Diagram
Draw a right triangle where:
- The height of the tower (180 m) is the vertical side (perpendicular).
- The horizontal distance from the foot of the tower to the cat is the base.
- The line of sight from the top of the tower to the cat forms the hypotenuse.

### Step 3: Identify the Angles
The angle of depression from the top of the tower to the cat is 30 degrees. In a right triangle, the angle of depression from the horizontal line at the top of the tower corresponds to the angle of elevation from the cat to the top of the tower. Therefore, the angle of elevation is also 30 degrees.

### Step 4: Use Trigonometric Ratios
In the right triangle formed, we can use the tangent function, which is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is the height of the tower (180 m), and the adjacent side is the distance from the foot of the tower to the cat (let's call it \(d\)).

### Step 5: Set Up the Equation
Using the tangent of the angle:
\[ \tan(30^\circ) = \frac{180}{d} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).

### Step 6: Solve for \(d\)
Substituting the value of \(\tan(30^\circ)\) into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{180}{d} \]
Cross-multiplying gives:
\[ d = 180 \cdot \sqrt{3} \]

### Step 7: Calculate the Value
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ d \approx 180 \cdot 1.732 \approx 311.76 \text{ m} \]

### Final Answer
The distance of the cat from the foot of the tower is approximately **311.76 meters**.

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