Question
Download Solution PDFComprehension
The top (M) of a tower is observed from three points P, Q and R lying in a horizontal straight line which passes directly along the foot (N) of the tower. The angles of elevations of M from P, Q and R are 30°, 45° and 60° respectively. Let PQ = a and QR = b
What is PN equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation:
The angles of elevation are:
From point P,\(\theta = 30^\circ \); from point Q, \(\theta = 45^\circ \); and from point R, \(\theta = 60^\circ \)
Using the tangent formula for each angle:
\( \tan(60^\circ) = \frac{h}{x} \implies h = \sqrt{3}x \)
\( \tan(45^\circ) = \frac{h}{QN} \implies h = QN\)
\( \tan(30^\circ) = \frac{h}{PN} \implies PN = h{\sqrt{3}} \)
from the figure PN = h + a
\(h+a = h\sqrt3\)
\(h = \frac{a}{\sqrt3 -1} \implies \frac{a \sqrt3 +1}{2}\)
= \((\frac{3 + \sqrt3}{2}) a\)
Hence, the correct answer is Option 2.
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