\section*{Problem 14}

A spy-plane circles point A at a distance \(10 \mathrm{~km}\) with speed \(1000 \mathrm{~km} / \mathrm{h}\). A missile is fired towards the plane from A at the same speed and moves so that it is always on the line between A and the plane. How long does it take to hit?


\section*{Solution}

Answer: 18 pi sec.

Let \(\mathrm{C}\) be the position of the spy-plane at the moment the missile is fired. Let \(\mathrm{B}\) be the point a quarter of the way around the circle from \(\mathrm{C}\) (in the direction the spy-plane is moving). Then the missile moves along the semi-circle on diameter \(\mathrm{AB}\) and hits the plane at \(\mathrm{B}\).

To see this take a point \(\mathrm{P}\) on the quarter circle and let the line \(\mathrm{AX}\) meet the semi-circle at \(\mathrm{Q}\). Let \(\mathrm{O}\) be the center of the semicircle. The angle BOQ is twice the angle BAQ, so the arc BP is the same length as the arc \(\mathrm{BQ}\). Hence also the arc \(\mathrm{AQ}\) is the same length as the arc \(\mathrm{CP}\).

