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 represent the statue, BC represent the pedestal, and D be the point on the ground from which elevation angles are measured.

In triangle BCD:

\(\frac{BC}{CD} = tan 45°\)

\(\frac{BC}{ CD} = 1 \)

\(BC = CD\)

In triangle ACD:

\(\frac{AB + BC}{ CD} = tan 60°\)

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

Given AB = 1.6:

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

\(1.6 + BC = BC \sqrt3\)

\(1.6 = BC \sqrt3 - BC\)

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

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

To rationalize the denominator:

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

\(BC = \frac{1.6 (\sqrt3 +1)}{ (\sqrt3)^2 - (1)^2}\)
\(BC = \frac{1.6 (\sqrt3 +1)}{3 - 1}\)

\(BC = \frac{1.6 (\sqrt3 +1)}{2}\)

\(BC = 0.8\, (\sqrt3 +1)\)

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