The angle of depression of a boat B from the top K of a cliff HK, 300 metres high is `30^(@)` . Find the distance of the boat from the foot H of the cliff.

Answer

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

### Step 1: Understand the Problem
We have a cliff that is 300 meters high, and we need to find the distance of a boat from the foot of the cliff. The angle of depression from the top of the cliff to the boat is 30 degrees.

### Step 2: Draw a Diagram
Draw a vertical line representing the cliff (HK) with a height of 300 meters. Mark point K at the top of the cliff and point H at the bottom. Draw a horizontal line from K to the right, representing the line of sight to the boat (B). The angle of depression from K to B is 30 degrees.

### Step 3: Identify the Right Triangle
In triangle BKH:
- The height of the cliff (KH) is the opposite side, which is 300 meters.
- The distance from the foot of the cliff to the boat (BH) is the adjacent side, which we need to find.

### Step 4: Use Trigonometric Ratios
Since we have the angle of depression (30 degrees), we can use the tangent function:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
Here, \(\theta = 30^\circ\), the opposite side is KH (300 meters), and the adjacent side is BH (which we will denote as x).

### Step 5: Set Up the Equation
Using the tangent function:
\[ \tan(30^\circ) = \frac{KH}{BH} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{300}{x} \]

### Step 6: Substitute the Value of \(\tan(30^\circ)\)
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
So, substituting this into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{300}{x} \]

### Step 7: Cross-Multiply to Solve for x
Cross-multiplying gives:
\[ x = 300 \sqrt{3} \]

### Step 8: Conclusion
Thus, the distance of the boat from the foot of the cliff (H) is:
\[ x = 300 \sqrt{3} \text{ meters} \]