A plane is flying 225 mph heading South 25°West. The wind is blowing South 80°East at 60 mph.
What is the bearing of the plane?
What is the smallest angle of the triangle?
What is the largest angle of the triangle?
What is the remaining angle of the triangle?
What is the ground speed of the plane?
Vp=225mi/h@(270-25) + 60mi/h@(270+80).
Vp = 225mi/h@245deg + 60mi/h@350deg.
Vp=(225cos245+60cos350) +
i(225sin245+60sin350).
Vp = (-95.1+59.1) + i(-203.9-10.4),
Vp = -36 - i214.3,
1. tan(Ar) = -214.3 / -36 = 5.95,
Ar = 80.5deg.
A = Ar + 180 = 80.5 + 180 = 260.5deg. =
Direction(bearing).
2. 90 - Ar = 90 - 80.5 = 9.5deg. = smallest angle.
3. 90deg = Largest angle.
4. 90 - 9.5 = 80.5deg.
5. Vp^2 = (-36)^2 + (214.3)^2 = 47,220.49,
Vp = 217.3mi/h.
To find the bearing of the plane, we need to add the headings of the plane and the wind. The plane is heading South 25° West, which means the bearing of the plane is 180° (South) + 25° West = 205°.
To find the smallest angle of the triangle, we can use the fact that the sum of the angles in a triangle is always 180°. The smallest angle of a triangle is opposite the smallest side. In this case, the side opposite the smallest angle is the side representing the wind direction, which is South 80° East. Therefore, the smallest angle is 80°.
To find the largest angle of the triangle, we know that the sum of the angles in a triangle is always 180°. The largest angle is opposite the largest side. In this case, the side representing the largest side is the ground speed of the plane. Therefore, the largest angle is 225°.
To find the remaining angle of the triangle, we can subtract the sum of the smallest and largest angles from 180°. Therefore, the remaining angle is 180° - (80° + 225°) = -125°.
However, angles cannot be negative, so we need to adjust it to a positive angle. We can add 360° to the -125° to get a positive angle measurement. Thus, the remaining angle of the triangle is 360° - 125° = 235°.
To find the ground speed of the plane, we can use the Law of Cosines. This law states that the square of the length of the side opposite an angle in a triangle is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those sides times the cosine of that angle.
In this case, we know two sides of the triangle: the wind speed (60 mph) and the plane speed (225 mph). We also know the angle between these two sides (80°). Using the Law of Cosines, we can calculate the ground speed of the plane:
Ground speed² = Wind speed² + Plane speed² - 2 * Wind speed * Plane speed * cos(angle)
Ground speed² = 60² + 225² - 2 * 60 * 225 * cos(80°)
Ground speed ≈ √(60² + 225² - 2 * 60 * 225 * cos(80°))
Using a calculator to calculate the expression, we find that the ground speed is approximately 251.13 mph.