To find the distance the plane traveled from point A to point B, we can apply the principles of trigonometry and use the Law of Sines or other suitable techniques, given the plane's altitude and the angles of elevation from both points.

Step-by-Step Solution:

1. Define variables:

   • The altitude of the plane: $h = 73757375$ feet (constant).
   • Angle of elevation at point A: $\theta_A = 16^\circ$.
   • Angle of elevation at point B: $\theta_B = 34^\circ$.
   • Let the horizontal distance from point A to the plane's perpendicular projection be $x_A$.
   • Let the horizontal distance from point B to the plane's perpendicular projection be $x_B$.
   • The distance traveled by the plane from point A to point B is $d = |x_B - x_A|$.

2. Use the tangent function: For each point, we can use the tangent of the angle of elevation to relate the altitude and the horizontal distance:

   $\tan(\theta_A) = \frac{h}{x_A}$ $\tan(\theta_B) = \frac{h}{x_B}$

3. Solve for $x_A$ and $x_B$: From the above equations:

   $x_A = \frac{h}{\tan(\theta_A)} = \frac{73757375}{\tan(16^\circ)}$ $x_B = \frac{h}{\tan(\theta_B)} = \frac{73757375}{\tan(34^\circ)}$

4. Calculate values for $x_A$ and $x_B$: Using the tangent values:

   • $\tan(16^\circ) \approx 0.2867$
   • $\tan(34^\circ) \approx 0.6682$

   Now calculate the horizontal distances:

   $x_A = \frac{73757375}{0.2867} \approx 257083495.6 \text{ feet}$ $x_B = \frac{73757375}{0.6682} \approx 110477163.1 \text{ feet}$

5. Find the distance traveled: The distance the plane traveled from point A to point B is the difference in horizontal distances:

   $d = |x_B - x_A| = |110477163.1 - 257083495.6| \approx 146606332.5 \text{ feet}$

Final Answer:

The distance the plane traveled from point A to point B is approximately 146,606,332.5 feet.

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Would you like more details on any of the steps, or do you have any questions? Here are some related questions for further practice:

1. How would the distance change if the altitude of the plane were different?
2. What is the formula to calculate the horizontal distance given an angle of elevation and height?
3. How can you apply trigonometric ratios in real-world scenarios involving radar and planes?
4. How can the Law of Sines be used to solve this type of problem instead of using tangents?
5. How do changes in the angles of elevation affect the total distance traveled?

Tip: When working with angles in elevation or depression, always make sure to use a calculator in the correct mode (degrees vs. radians) to avoid calculation errors.