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To solve this problem, we can use trigonometry. Here's the step-by-step solution:

1. **Label the Points and Angles**:
   - Let the plane be at point \( P \).
   - Ships are at points \( A \) and \( B \), with \( AB = 9 \) miles.
   - The angles of depression from \( P \) to \( A \) and \( B \) are \( 49^\circ \) and \( 69^\circ \) respectively.

2. **Translate the Angles of Depression to Angles of Elevation**:
   - Since the angles of depression to the ships are given from the plane, we can assume that the angles of elevation from the ships to the plane are the same (due to the parallel lines concept).
   - Therefore:
     - The angle at \( A \) (angle \( \angle PAB \)) is \( 49^\circ \).
     - The angle at \( B \) (angle \( \angle PBA \)) is \( 69^\circ \).

3. **Calculate the Angle at \( P \)**:
   - The sum of angles in a triangle is \( 180^\circ \).
   - So, the angle \( \angle APB \) at \( P \) is:
     \[
     \angle APB = 180^\circ - 49^\circ - 69^\circ = 62^\circ
     \]

4. **Apply the Law of Sines**:
   - We want to find the distance from the plane to ship \( A \), which we'll call \( PA \).
   - Using the Law of Sines:
     \[
     \frac{PA}{\sin(69^\circ)} = \frac{9}{\sin(62^\circ)}
     \]
   - Rearranging to solve for \( PA \):
     \[
     PA = \frac{9 \cdot \sin(69^\circ)}{\sin(62^\circ)}
     \]

5. **Calculate the Value**:
   - Using approximations for sine values:
     - \( \sin(69^\circ) \approx 0.9336 \)
     - \( \sin(62^\circ) \approx 0.8829 \)
   - Plugging in:
     \[
     PA = \frac{9 \cdot 0.9336}{0.8829} \approx \frac{8.4024}{0.8829} \approx 9.5 \text{ miles}
     \]

**Answer**: The distance from the plane to the ship at point \( A \) is approximately **9.5 miles**.

Would you like more details on any part of this solution, or have any questions?

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### Related Questions:
1. How would the solution change if the distance between ships \( A \) and \( B \) were different?
2. What trigonometric concepts are essential for solving this type of problem?
3. How can we verify our answer using a different trigonometric method?
4. How would we calculate the distance to ship \( B \) instead of \( A \)?
5. What other real-world applications might involve using the Law of Sines?

### Tip:
Remember, for any problem involving angles of depression or elevation, carefully translate these to angles of elevation or depression at ground level to set up the triangle correctly.

A pilot is flying over the ocean. She determines that the angles of depression to two ships are 69° and 49°, as shown in the figure below. The ships are 9 miles apart. Find the distance from the plane to the ship at point A. Round your answer to the nearest tenth of a mile.

Learn how to calculate the distance from a plane to a ship using angles of depression and the Law of Sines. In this problem, a pilot observes two ships that are 9 miles apart with angles of depression of 69° and 49°. The solution involves trigonometric concepts and provides a step-by-step approach for students in Grades 10-12.

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