Question

A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Answer 1

Let AB be the statue, BC be the pedestal, and D be the point on the ground from where the elevation angles are to be measured. 

In ∆BCD,

\(\frac{BC}{CD} = tan 45°\)

\(\frac{BC}{ CD} = 1 \)

\(BC = CD\)

In ∆ACD,

\(\frac{AB + BC}{ BC} = tan 60°\)

\(\frac{AB + BC }{ BC} = \sqrt3\)

\(1.6 + BC = BC \sqrt3\)

\(BC = (\sqrt3 -1) = 1.6\)

\(BC =\frac{ (1.6) (\sqrt3 +1)}{ (\sqrt3 -1) (\sqrt3+ 1)}\)

\(BC = \frac{1.6 (\sqrt3+1)}{ (\sqrt3)^2 - (1)^2}\)
\(BC = \frac{1.6 (\sqrt3 +1)}2 = 0.8\, (\sqrt3 +1)\)

Therefore, the height of the pedestal is\(0.8\, (\sqrt3 +1)\) m.