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...
Question 1 Report
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.
Answer Details
Let the pilot be at a height \(H\) above the ground, with the horizontal distance to the tower equal to 8.5 km. The angle of depression to the base is \(33^\circ\) and to the top is \(30^\circ\) (the top is higher, so its depression is smaller).
Height of pilot above the ground, using the base:
\[\tan 33^\circ = \frac{H}{8.5} \Rightarrow H = 8.5\tan 33^\circ = 8.5(0.6494) = 5.520\text{ km}.\]
(a) Shortest distance from the pilot to the base. This is the straight line (line of sight):
\[d = \frac{8.5}{\cos 33^\circ} = \frac{8.5}{0.8387} = 10.135\text{ km} \approx \mathbf{10.1\text{ km}}.\]
(b) Height of the control tower. The height of the pilot above the top of the tower is
\[8.5\tan 30^\circ = 8.5(0.5774) = 4.908\text{ km}.\]
Therefore the tower height is
\[H - 8.5\tan 30^\circ = 5.520 - 4.908 = 0.612\text{ km}.\]
The height of the control tower \(\approx \mathbf{0.612\text{ km}}\) (about 612 m), correct to three significant figures.
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