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To solve the problem, we use the Law of Cosines. Given:

- Distance between Station \( A \) and the plane: \( 7.4 \) miles
- Distance between Station \( B \) and the plane: \( 6.9 \) miles
- Distance between Stations \( A \) and \( B \): \( 2.4 \) miles

We are asked to find the angle of elevation from Station \( A \) to the plane. Let's call this angle \( \theta \).

### Law of Cosines
The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab\cos(C)
\]
where:
- \( c \) is the side opposite the angle \( C \).
- \( a \) and \( b \) are the other two sides.

### Applying the Law of Cosines
Let:
- \( a = 6.9 \) miles (distance from Station \( B \) to the plane)
- \( b = 2.4 \) miles (distance between the stations)
- \( c = 7.4 \) miles (distance from Station \( A \) to the plane)

Plug these into the formula:
\[
(7.4)^2 = (6.9)^2 + (2.4)^2 - 2(6.9)(2.4)\cos(\theta)
\]
\[
54.76 = 47.61 + 5.76 - 33.12\cos(\theta)
\]
\[
54.76 - 53.37 = -33.12\cos(\theta)
\]
\[
1.39 = -33.12\cos(\theta)
\]
\[
\cos(\theta) = \frac{1.39}{-33.12} \approx -0.04196
\]

Now, calculate \( \theta \):
\[
\theta = \cos^{-1}(-0.04196) \approx 92.4^\circ
\]

This suggests the angle is roughly \( 92^\circ \). If there are any discrepancies, they might be due to rounding or interpretation differences in the given answer.

Would you like further details or have any questions?

### Additional Questions
1. How would the Law of Sines apply to this problem if you had an additional angle?
2. Can you calculate the angle of elevation from Station \( B \) instead?
3. What if the distance between the stations was longer, say \( 5 \) miles—how would it change the angle?
4. How do rounding errors impact the accuracy of trigonometric calculations?
5. Could you determine the height of the airplane above ground using this information?

### Tip
Always double-check your calculations when applying the Law of Cosines, especially when working with angles close to \( 90^\circ \), as small changes can significantly affect the result.

Two radar stations 2.4 miles apart are tracking an airplane. The straight-line distance between Station A and the plane is 7.4 miles. The straight-line distance between Station B and the plane is 6.9 miles. What is the angle of elevation from Station A to the plane? Round to the nearest degree.

Learn how to find the angle of elevation from Station A to an airplane tracked by two radar stations using the Law of Cosines. The distances between the airplane and the stations are 7.4 miles and 6.9 miles, with a distance of 2.4 miles between the stations. Suitable for high school students studying trigonometry.

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