An airplane is flying at an elevation level of 6200 feet above a straight highway. The pilot spots two cars on the highway in front of him. The angle of depression to one car is 25 and the angle of depression to the other car is 17. How far apart are the cars?

To solve this, we can use trigonometry and the concept of angles of depression to find the distances from the plane to each car on the highway, then calculate how far apart they are.

### Step 1: Define the problem with right triangles
- The airplane is 6200 feet above the highway.
- Let \( A \) and \( B \) represent the positions of the two cars on the highway, and let \( P \) represent the airplane's position.
- The angle of depression from the airplane to car \( A \) is 25°, and to car \( B \) is 17°.
- We want to find the horizontal distance between cars \( A \) and \( B \).

### Step 2: Set up equations using tangent
In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Let \( d_A \) be the horizontal distance from the plane to car \( A \), and \( d_B \) be the horizontal distance from the plane to car \( B \).

For car \( A \):
\[
\tan(25^\circ) = \frac{6200}{d_A}
\]
Rearrange to solve for \( d_A \):
\[
d_A = \frac{6200}{\tan(25^\circ)}
\]

For car \( B \):
\[
\tan(17^\circ) = \frac{6200}{d_B}
\]
Rearrange to solve for \( d_B \):
\[
d_B = \frac{6200}{\tan(17^\circ)}
\]

### Step 3: Calculate \( d_A \) and \( d_B \)

Using approximate values:
\[
d_A \approx \frac{6200}{0.4663} \approx 13298.96 \text{ feet}
\]
\[
d_B \approx \frac{6200}{0.3057} \approx 20280.86 \text{ feet}
\]

### Step 4: Find the distance between the cars
The distance \( D \) between cars \( A \) and \( B \) is the difference between \( d_B \) and \( d_A \):
\[
D = d_B - d_A
\]
\[
D \approx 20280.86 - 13298.96 = 6981.9 \text{ feet}
\]

### Final Answer
The two cars are approximately **6982 feet apart**.

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

---

1. What is the effect of changing the airplane's altitude on the distance calculation?
2. How would the answer change if the angles of depression were swapped?
3. Can this approach be applied if the highway were not straight?
4. What is an angle of depression, and how does it relate to an angle of elevation?
5. How can you verify this result with a drawing or another method?

**Tip:** To solve similar problems, break them down by identifying right triangles and using trigonometric functions like tangent for distances.

Explore how to calculate the distance between two cars on a highway based on angles of depression from an airplane flying at 6200 feet. This detailed solution demonstrates the use of trigonometry to find distances, resulting in cars being approximately 6982 feet apart.

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