From a horizontal distance of 8.5 km, a pilot... | WAEC SSCE - General Mathematics - 1991 (Essay)

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

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.

Read lesson note on Angles Of Elevation And Depression (WAEC)

Read lesson note on Fractions, Decimals, Approximations And Percentage (JAMB)

Answer Details

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.

Read lesson note on Angles Of Elevation And Depression (WAEC)

Read lesson note on Fractions, Decimals, Approximations And Percentage (JAMB)

Angles Of Elevation And Depression

Fractions, Decimals, Approximations And Percentage

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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...

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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...

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