Angles of elevation and depression Problem set A Worked solutions

A lamp post is seen from a point 18 m from its foot. The angle of elevation of the top of the lamp post is 32°, and the observer's eyes are 1.7 m above the ground. Find the height of the lamp post.

• \(\tan 32^\circ = \dfrac{h}{18}\), \(h = 11.25\)
• Add the eye height: 11.25 + 1.7 = 12.9 m

From the top of a cliff 45 m above the sea, the angle of depression of a boat is 23°. Find the horizontal distance of the boat from the foot of the cliff.

• The angle of depression from the top equals the angle of elevation from the boat, so the angle at the boat is 23°.
• \(\tan 23^\circ = \dfrac{45}{d}\), \(d = 106.0\) m

From a point A the angle of elevation of the top of a flagpole is 42°. From a point 25 m further from the flagpole, on the same level and in line with it, the angle of elevation is 27°. The observer's eyes are 1.5 m above the ground. Find the height of the flagpole.

• \(\angle ABT = 27^\circ\), \(\angle BAT = 180^\circ - 42^\circ = 138^\circ\), \(\angle ATB = 42^\circ - 27^\circ = 15^\circ\)
• \(\dfrac{AT}{\sin 27^\circ} = \dfrac{25}{\sin 15^\circ}\), so \(AT = 43.85\)
• \(h = AT \sin 42^\circ = 43.85 \times \sin 42^\circ = 29.34\)
• Add the eye height: 29.34 + 1.5 = 30.8

The angle of elevation of the top of a church spire from a point A is 31°. Anil walks 60 m directly away from the spire to a point B, where the angle of elevation is 19°. His eyes are 1.7 m above the ground. Find the height of the spire.

• \(\angle ABT = 19^\circ\), \(\angle BAT = 180^\circ - 31^\circ = 149^\circ\), \(\angle ATB = 31^\circ - 19^\circ = 12^\circ\)
• \(\dfrac{AT}{\sin 19^\circ} = \dfrac{60}{\sin 12^\circ}\), so \(AT = 93.95\)
• \(h = AT \sin 31^\circ = 93.95 \times \sin 31^\circ = 48.39\)
• Add the eye height: 48.39 + 1.7 = 50.1

From the top of a cliff the angle of depression of a boat is 15°. The boat sails 200 m directly towards the cliff and the angle of depression is then 28°. Find the height of the top of the cliff above the sea.

• \(\angle ABT = 15^\circ\), \(\angle BAT = 180^\circ - 28^\circ = 152^\circ\), \(\angle ATB = 28^\circ - 15^\circ = 13^\circ\)
• \(\dfrac{AT}{\sin 15^\circ} = \dfrac{200}{\sin 13^\circ}\), so \(AT = 230.1\)
• \(h = AT \sin 28^\circ = 230.1 \times \sin 28^\circ = 108.0\)

The top of a flagpole is 12 m above an observer's eye level. The angle of elevation of the top is 40° from one point. The observer walks directly away from the flagpole, and the angle of elevation of the top is then 25°. How far does the observer walk?

• \(\text{First distance} = \dfrac{12}{\tan 40^\circ} = 14.30\)
• \(\text{Second distance} = \dfrac{12}{\tan 25^\circ} = 25.73\)
• Distance walked: \(25.73 - 14.30 = 11.4\) m