74. Test Flight. A new type of aircraft took off from Oslo, heading for the shortest route to airport $X$, located in South America, right on the equator. Witnesses to the departure from Oslo see the aircraft disappear on the horizon at a point lying exactly to the west.

What is the length of the flight path? At what point on the horizon should the spectators waiting for the aircraft at airport $X$ be looking?
74. The shortest path on the Earth (sphere) is the arc of a great circle, i.e.
![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-148.jpg?height=358&width=738&top_left_y=1334&top_left_x=958)

Fig. 155. circle formed by the intersection of the sphere with a plane passing through the center of the sphere.

Since the plane, taking off from Oslo, disappeared from the eyes of the spectators at the airport at a point on the horizon lying exactly to the west, the plane flew along a great circle, intersecting the meridian at Oslo ( $\alpha=10^{\circ} 43^{\prime}$ east longitude) at a right angle.

Let in Fig. $156 S$ and $O$ denote the center of the Earth and the point where Oslo is located, respectively, and let $Q, G, O^{\prime}$ denote the points of intersection with the equator: the great circle along which the plane flew, the zero meridian, and the meridian passing through Oslo, respectively.

It is easy to notice that the angle $Q S O^{\prime}$ is a right angle; hence we have:

$$
\angle G S Q=90^{\circ}-\angle G S O^{\prime}=90^{\circ}-10^{\circ} 43^{\prime}=79^{\circ} 17^{\prime}
$$

Therefore, the point where the plane should land lies on the equator at $79^{\circ} 17^{\prime}$ west longitude (the point with these coordinates lies in the province of Pichincha, 120 km west of Quito, the capital of Ecuador), and since the arc $O Q$ is $1 / 4$ of a great circle, the flight path is approximately 10,000 km.

The plane $O S Q$ is inclined to the plane of the equator at an angle $\beta=59^{\circ} 55^{\prime}$; at this angle relative to the east direction, the spectators at the airport in Ecuador should wait for the plane.

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-149.jpg?height=751&width=740&top_left_y=547&top_left_x=358)

Fig. 156.