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To solve for the height of the cliff in this problem, we will use trigonometry based on the angle of depression and the information given in the diagram.

### Step-by-step approach:

1. **Given information:**
   - The distance between the base of the cliff and the ship (horizontal distance) is 650 m.
   - The angle of depression is \(43^\circ\).
   - Tony's height is 1.82 m.
   
2. **Understanding the angle of depression:**
   - The angle of depression is the angle between the horizontal line at Tony's eye level and the line of sight to the ship. In this case, it is \(43^\circ\).
   
3. **Triangle Setup:**
   - From the diagram, we can see that this forms a right triangle, where:
     - The horizontal distance (adjacent) is 650 m.
     - The vertical distance (opposite), which we need to find, is the height of the cliff minus Tony's height.

4. **Using the tangent function:**
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
   \]
   where:
   - \(\theta = 43^\circ\),
   - opposite is the height of the cliff minus Tony's height,
   - adjacent is the horizontal distance (650 m).

   Substituting values:
   \[
   \tan(43^\circ) = \frac{h_{\text{cliff}} - 1.82}{650}
   \]
   
   Solving for \(h_{\text{cliff}}\):
   \[
   h_{\text{cliff}} - 1.82 = 650 \times \tan(43^\circ)
   \]

5. **Calculating:**
   Let's compute the height of the cliff.The height of the cliff is approximately 608 meters when rounded to the nearest meter.

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:
1. How would the height of the cliff change if Tony's height were different?
2. What happens to the height if the angle of depression decreases?
3. How would you solve this problem if the horizontal distance were unknown but the height was given?
4. How does the tangent function relate to the geometry of this situation?
5. What other trigonometric functions could be used to solve similar problems?

**Tip:** When solving for unknowns in right triangles, it's essential to correctly identify which side corresponds to the adjacent, opposite, and hypotenuse based on the angle you're working with.

Tony standing at the top of a cliff sights a ship 650 m at sea. The angle of depression of the ship from Tony is 43°. If Tony is 1.82 m tall, find the height of the cliff to the nearest meter.

Solve a real-world problem involving trigonometry by calculating the height of a cliff. Given a 650 m horizontal distance, a 43° angle of depression, and a person's height of 1.82 m, use the tangent function to determine the cliff's height. Suitable for students in Grades 9-12.

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