(a) A tower and a building stand on the same horizontal level. From the point P at the bottom of the building, the angle of elevation of the top, T of the tower is 65^\circ. From the top Q of the building, the angle of elevation of the point T is 25^\circ. If the building is 20m high, calculate the distance PT.
(b) Hence or otherwise, calculate the height of the tower. [Give your answers correct to 3 significant figures].
Verified Correct Answer: .
Step-by-Step Mathematical Solution
(a)
\(< QPB = 90^\circ - 65^\circ = 25^\circ\)
Also, \(< TQP = 90^\circ + 25^\circ = 115^\circ\)
\(\therefore < QTP = 180^\circ - (25^\circ + 115^\circ) = 40^\circ\)
Using sine rule, \(\frac{PQ}{\sin 40^\circ} = \frac{PT}{\sin 115^\circ}\)
\(PT = \frac{20 \sin 115^\circ}{\sin 40^\circ} = \frac{20 \sin 65^\circ}{\sin 40^\circ}\)
= \(28.199m \approxeq 28.2m\) (to 3 sig. fig)
(b) Height of the tower = ST
\(\sin 65 = \frac{ST}{PT} = \frac{ST}{28.199}\)
\(ST = 28.199 \times \sin 65^\circ = 25.557m\)
\(\approxeq 25.6m\) (to 3 sig. figs)
Source: Myschool Past Questions Archive (Record #54551)