From a horizontal distance of 8.5 km, a pilot observes that the angles of depression of the top and the base of a control tower are 30° and 33° respectively. Calculate, correct to three significant figures :
(a) the shortest distance between the pilot and the base of the control tower;
(b) the height of the control tower.
Explanation
Let T and B be the top and base of the control tower respectively; P, the point of pilot’s observation and PS, the line of observation.
(a) Shortest distance is |PB|.
(therefore cos 33° = frac{8.5}{PB})
(PB = frac{8.5}{cos 33} = frac{8.5}{0.8387})
(|PB| = 10.135km approxeq 10.1km)
(b) Height of the control tower = |SB| – |ST|
(therefore tan 33° = frac{|SB|}{8.5 km} implies |SB| = 8.5 tan 33°)
(|SB| = 5.52 km)
(tan 30° = frac{|ST|}{8.5 km} implies |ST| = 8.5 tan 30°)
(|ST| = 4.907 km)
(therefore) The height of the tower = 5.52 km – 4.907 km = 0.613 km.