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To solve the problem of determining the distance of the buoy from the foot of a 130 m high cliff given the angle of depression is 18°, we can use trigonometry. Specifically, we apply the tangent function since it relates the angle to the opposite side (height of the cliff) and adjacent side (distance from the cliff's base to the buoy).

Here’s the setup:
1. **Angle of depression** is 18°. Since the angle is measured from the horizontal looking down, the corresponding angle from the foot of the cliff looking up to the top is also 18° due to alternate interior angles.
2. **Height of the cliff (opposite side)**: 130 m.
3. **Adjacent side (x)**: This is what we need to find.

Using the tangent function:
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]

Plugging in the values:
\[ \tan(18^\circ) = \frac{130}{x} \]

To find \(x\), the distance from the foot of the cliff to the buoy, we rearrange the equation:
\[ x = \frac{130}{\tan(18^\circ)} \]

Let's calculate that now.The distance of the buoy from the foot of the cliff is approximately 400.1 meters.

Would you like any more details or have any other questions on this topic? Here are some related questions you might consider:

1. How would the distance change if the angle of depression were larger or smaller?
2. What would be the angle of depression if the buoy were twice as far from the cliff?
3. How would you determine the distance to the buoy if given only the angle of elevation from the base of the cliff?
4. What are the practical applications of using angles of depression and elevation in real-world scenarios?
5. Can you solve for the height of the cliff if the angle and distance to the buoy are known?

**Tip:** When working with trigonometric functions in calculations, always ensure your calculator is set to the correct mode (degrees or radians) based on the units used for the angles in your problem.

From the top of a vertical cliff 130 m high, the angle of depression of a buoy at sea is 18°. What is the distance of the buoy from the foot of the cliff?

Learn how to calculate the distance from the foot of a cliff to a buoy at sea using trigonometric functions. Given the angle of depression (18°) and the height of the cliff (130 m), this step-by-step guide explains how to apply the tangent function to find the distance. Suitable for students in Grades 9-11.

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