From the top of a vertical cliff 125 m high, the angle of depression of a boat is 18.833o. Find the distance of the boat from the cliff.

To solve this problem, we can use trigonometry. Specifically, we’ll use the tangent function, which relates the angle of depression to the opposite side (the height of the cliff) and the adjacent side (the distance of the boat from the cliff).

### Given:
- Height of the cliff (\(h\)) = 125 m
- Angle of depression (\(\theta\)) = 18.833°

### To find:
- The distance of the boat from the cliff (\(d\))

### Using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\]

Rearranging to solve for \(d\):
\[
d = \frac{h}{\tan(\theta)}
\]

### Plugging in the values:
\[
d = \frac{125}{\tan(18.833^\circ)}
\]

Now, calculate the value:

\[
\tan(18.833^\circ) \approx 0.3404
\]

\[
d = \frac{125}{0.3404} \approx 367.07 \text{ meters}
\]

### Final Answer:
The distance of the boat from the cliff is approximately **367.07 meters**.

Would you like further details or have any questions? 

Here are five related questions to further explore the concept:

1. What would be the distance of the boat if the cliff were 200 meters high instead of 125 meters?
2. How would the distance change if the angle of depression were 25° instead of 18.833°?
3. Can you calculate the horizontal distance if the angle of depression is given and the height of the cliff is unknown?
4. What role does the angle of elevation play in this kind of trigonometric problem?
5. How would you find the height of the cliff if the distance of the boat and the angle of depression are given?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) when solving trigonometric problems.

Learn how to find the distance of a boat from a cliff using trigonometry. Given the height of the cliff (125 m) and the angle of depression (18.833°), calculate the distance using the tangent function. Suitable for Grades 10-12.

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