From the top of a cliff 92 m high. the angle of depression of a buoy is `20^(@)` . Calculate, to the nearest metre, the distance of the buoy from the foot of the cliff

Answer

To solve the problem, we will use the concept of angles of depression and trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a cliff that is 92 meters high, and we need to find the horizontal distance from the foot of the cliff to a buoy, given that the angle of depression to the buoy is 20 degrees.

### Step 2: Draw a Diagram
1. Draw a vertical line representing the cliff (AB) with a height of 92 meters.
2. Mark point C at the base of the cliff (foot of the cliff).
3. Mark point D where the buoy is located.
4. The angle of depression from point A (top of the cliff) to point D (the buoy) is 20 degrees.

### Step 3: Identify the Right Triangle
From the diagram, we can see that triangle ABC is a right triangle where:
- AB = height of the cliff = 92 m
- Angle ACB = 20 degrees (angle of depression)
- BC = distance from the foot of the cliff (C) to the buoy (D), which we need to find.

### Step 4: Use the Tangent Function
In triangle ABC, we can use the tangent function, which relates the opposite side to the adjacent side:
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is AB (the height of the cliff), and the adjacent side is BC (the distance we want to find).

Thus, we can write:
\[ \tan(20^\circ) = \frac{AB}{BC} \]
Substituting the known values:
\[ \tan(20^\circ) = \frac{92}{BC} \]

### Step 5: Rearranging the Equation
Rearranging the equation to solve for BC:
\[ BC = \frac{92}{\tan(20^\circ)} \]

### Step 6: Calculate the Value of BC
Using a calculator, we find:
\[ \tan(20^\circ) \approx 0.364 \]
Now substituting this value into the equation:
\[ BC = \frac{92}{0.364} \approx 253.3 \]

### Step 7: Round to the Nearest Metre
Rounding 253.3 to the nearest metre gives us:
\[ BC \approx 253 \text{ m} \]

### Final Answer
The distance of the buoy from the foot of the cliff is approximately **253 meters**.

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