South African Mathematics Olympiad First Round

Question

At a height of *h* metres above sea level, the distance d, in kilometres, to the visible horizon is given approximately by the formula d = 8h5. A mountaineer on the side of Table Mountain can see a distance of 80 kilometres to the horizon. How high, in metres, is the mountaineer above sea level? (A) 1 000 (B) 500 (C) 650 (D) 250 (E) 400

Accepted Answer

Since d = 8h5, it follows that d^2 = 64h5, so h = 564d^2. With d = 80, this gives d^2 = 6400, so h = 564 6400 = 500.