The island of Gammaland is totally flat and subject to a constant wind of \(w\) kh\(^{-1},\) blowing from the West. Its southernmost shore stretches almost indefinitely, due east and west, from the coastal city of Alphabet. A novice pilot is making her first solo flight from Alphaport to the town of Betaville which lies north-east of Alphaport. Her instructor has given her the correct heading to reach Betaville, flying at the plane's recommended airspeed of \(v\) kh\(^{-1},\) where \(v>w.\) On reaching Betaport the pilot returns with the opposite heading to that of the outward flight and, so featureless is Gammaland, that she only realises her error as she crosses the coast with Alphaport nowhere in sight. Assuming that she then turns West along the coast, and that her outward flight took \(t\) hours, show that her return flight takes \[ \left(\frac{v+w}{v-w}\right)t\ \text{hours.} \] If Betaville is \(d\) kilometres from Alphaport, show that, with the correct heading, the return flight should have taken \[ t+\frac{\sqrt{2}wd}{v^{2}-w^{2}}\ \text{hours.} \]