Let's break this down step-by-step to find how far the airplane is from the starting point (point A).
1. Understanding the Problem:
- The airplane flies 120 miles from point A at an angle of 110°.
- Then, it changes direction and flies 70 miles at an angle of 255°.
- We need to compute the distance from the final position back to the starting point A.
2. Breaking Down into Components:
We will split the movement into horizontal (x) and vertical (y) components based on trigonometry principles:
- Distance 1: 120 miles at 110°
- [tex]\( x_1 = 120 \cos(110°) \)[/tex]
- [tex]\( y_1 = 120 \sin(110°) \)[/tex]
- Distance 2: 70 miles at 255°
- [tex]\( x_2 = 70 \cos(255°) \)[/tex]
- [tex]\( y_2 = 70 \sin(255°) \)[/tex]
3. Calculating Total Displacement Components:
Summing the individual components to get the total displacement:
- Total horizontal displacement, [tex]\( x_{total} = x_1 + x_2 \)[/tex]
- Total vertical displacement, [tex]\( y_{total} = y_1 + y_2 \)[/tex]
4. Resulting Components:
The x and y components after each segment of the trip are:
- [tex]\( x_{total} \approx -59.16 \)[/tex]
- [tex]\( y_{total} \approx 45.15 \)[/tex]
5. Computing the Overall Distance:
The straight-line distance (displacement) back to point A can be calculated using the Pythagorean theorem:
[tex]\[
D = \sqrt{x_{total}^2 + y_{total}^2}
\][/tex]
Substituting the values:
[tex]\[
D = \sqrt{(-59.16)^2 + (45.15)^2}
\][/tex]
6. Final Calculation:
- The calculated distance is approximately: [tex]\( D \approx 74 \)[/tex] miles.
Thus, the airplane is approximately 74 miles from point A.
