An airplane is 35 km north and 55 km east of an airport. It is 7 km up in the air. What is the angle along the ground that the airplane is at, as a direction east of north?
65.4°
57.5°
58.7°
52.4°
To find the angle along the ground that the airplane is at, we can use trigonometry.
We can form a right triangle with the airplane's position relative to the airport. The horizontal distance from the airport to the airplane is 55 km (east), and the vertical distance is 35 km (north). The hypotenuse of this right triangle represents the ground distance between the airplane and the airport.
Using the Pythagorean theorem, we can find the hypotenuse:
c^2 = a^2 + b^2
c^2 = (55 km)^2 + (35 km)^2
c^2 = 3025 km^2 + 1225 km^2
c^2 = 4250 km^2
c = √(4250 km^2)
c ≈ 65.19 km
Now that we have the horizontal and vertical distances, we can use the tangent function to find the angle:
tan(angle) = opposite/adjacent
tan(angle) = 35 km/55 km
angle = arctan(35/55)
angle ≈ 32.5°
However, this gives us the angle north of east. To get the angle east of north, we subtract this angle from 90°:
90° - 32.5° ≈ 57.5°
Therefore, the angle along the ground that the airplane is at, as a direction east of north, is approximately 57.5°.
The closest option is 57.5°.