Math Problem Statement
Fatoumata spots an airplane on radar that is currently approaching in a straight line, and that will fly directly overhead. The plane maintains a constant altitude of 6875 feet. Fatoumata initially measures an angle of elevation of 17 degrees to the plane at point A. At some later time, she measures an angle of elevation of 40 degrees to the plane at point B. Find the distance the plane traveled from point A to point B. Round your answer to the nearest foot if necessary.
Solution
To solve this problem, we can use trigonometry, particularly the tangent function, which relates the angle of elevation, the altitude of the airplane, and the horizontal distance from the observer to the point directly beneath the airplane.
Step 1: Set up the problem
- Let feet be the altitude of the plane.
- Let be the horizontal distance from point A to the point directly under the airplane when the angle of elevation is 17 degrees.
- Let be the horizontal distance from point B to the point directly under the airplane when the angle of elevation is 40 degrees.
- Let be the distance the plane traveled from point A to point B.
We need to find .
Step 2: Use the tangent function
For point A:
For point B:
Step 3: Calculate the values of and
First, find the values of and :
Now, calculate and :
Step 4: Find the distance the plane traveled
Final Answer
The distance the plane traveled from point A to point B is approximately 14,298 feet.
Would you like more details or have any questions?
Related Questions:
- How would the distance change if the plane’s altitude were different?
- What if the angles of elevation were measured in radians instead of degrees?
- How would you calculate the distance traveled if the angle of elevation at point B were less than at point A?
- Can you derive a general formula for the distance traveled based on the angles and altitude?
- What happens to the distance if one of the angles of elevation is 90 degrees?
Tip: In problems involving angles and distances, drawing a diagram often helps in understanding the relationships between different components.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Angles of Elevation
Right Triangle Geometry
Formulas
Tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
Theorems
-
Suitable Grade Level
Grades 10-12
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