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.