A statue 1.46 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°. The height of the pedestal (round the answer up to one decimal place) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2.0 m
In a right triangle, \(\tan\) of an angle = opposite side ÷ adjacent side. Let the pedestal OB be \(h\) m and P the point on the ground; the top of the statue A is then \(h+1.46\) m above the ground. Both angles are seen from P, so both triangles share the base PO.
The distance PO (from the 45° angle to the top of the pedestal):
In triangle POB, \(\tan45^\circ=\displaystyle\frac{h}{PO}=1\), so the base PO is also \(h\)
The height \(h\) (from the 60° angle to the top of the statue):
In triangle POA, \(\tan60^\circ=\displaystyle\frac{h+1.46}{h}=\sqrt{3}\)
\(\sqrt{3}\,h-h=1.46\), so \((\sqrt{3}-1)\,h=1.46\)
\(h=\displaystyle\frac{1.46}{0.732}\approx1.99\), which rounds to 2.0 m