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 ...
Question 1 Report
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°.
Answer Details
\(H\) is 20 m from the foot of the tower. From \(H\), \(P\) has elevation \(30^\circ\) and the top \(T\) has elevation \(50^\circ\).
Heights above ground of the two points:
\[\text{height of }P = 20\tan 30^\circ = 20(0.5774) = 11.547\text{ m}\]
\[\text{height of }T = 20\tan 50^\circ = 20(1.1918) = 23.835\text{ m}\]
(a) \(|PT| = 23.835 - 11.547 = 12.288 \approx 12.3\text{ m (3 s.f.)}\).
(b) Distance \(|HT|\) (the slant line to the top):
\[|HT| = \frac{20}{\cos 50^\circ} = \frac{20}{0.6428} = 31.1\text{ m (3 s.f.)}\]
(c) For the angle of depression of \(H\) from the top to be \(40^\circ\), the angle of elevation of \(T\) from \(H\) must be \(40^\circ\). With the top still \(23.835\) m high, the required horizontal distance \(x\) satisfies:
\[\tan 40^\circ = \frac{23.835}{x} \;\Rightarrow\; x = \frac{23.835}{0.8391} = 28.4\text{ m (3 s.f.)}\]
So \(H\) must be \(28.4\) m from the foot of the tower.
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