A point H is 20 m away from the foot of a tower on the same horizontal ground. From the point H, the angle of elevation of the point P on the tower and the top (T) of the tower are 30° and 50° respectively. Calculate, correct to 3 significant figures :
(a) /PT/; (b) the distance between H and the top of the tower
(c) The position of H if the angle of depression of H from the top of the tower is to be 40°.
Explanation
(a) \(\tan 50° = \frac{|TF|}{20} \implies |TF| = 20 \tan 50\)
= \(20 \times 1.1918\)
= \(23.836 m\)
\(\tan 30 = \frac{|PF|}{20} \implies |PF| = 20 \tan 30\)
= \(20 \times 0.5774\)
= \(11.548 m\)
\(\therefore |PT| = 23.836 m - 11.548m\)
= \(12.288 m \approxeq 12.3 m\)
(b) \(|TH| = \frac{20}{\cos 50}\)
= \(\frac{20}{0.6428}\)
= \(31.11 m\)
(c) \(|HF| = \frac{20 \tan 50°}{\tan 40°}\)
= \(\frac{20 \times 1.1918}{0.8391}\)
= \(28.41 m\)