Question

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^o} $ and from the same point the angle of elevation of the top of the pedestal is $ {45^o} $. Find the height of the pedestal [Use$ \sqrt 3 = 1.73 $]

Answer 1

Hint: The problem can be solved by using trigonometry. The formula for the tangent of the angle should be used 2 times. Tangent of an angle is given by, $ \tan \theta = \dfrac{P}{B} $, where P = perpendicular and B = base of the right triangle.

Complete step-by-step answer:
The figure according to the condition given in the question is as follows,

In right triangle, DBC right angled at C
$ \tan {45^o} = \dfrac{P}{B} = \dfrac{h}{{BC}} $
The value of $ \tan {45^o} = 1 $
$
   \Rightarrow \dfrac{h}{{BC}} = 1 \\
  h = BC......(1) \\
 $



In right triangle, ABC right angled at C
$
  \tan {60^o} = \dfrac{P}{B} \\
  \tan {60^o} = \dfrac{{AC}}{{BC}} \\
  \tan {60^o} = \dfrac{{h + 1.46}}{{BC}} \\
 $

The value of $ \tan {60^o} = \sqrt 3 $
$ \Rightarrow \dfrac{{h + 1.46}}{{BC}} = \sqrt 3 $
$ \dfrac{{h + 1.46}}{h} = \sqrt 3 ......(2) $

Substitute h = BC in equation (2)
$ h + 1.46 = h\sqrt 3 ......(3) $

The value of $ \sqrt 3 = 1.73 $ as per the question, substitute it in equation (3)
$
  h + 1.46 = 1.73h \\
  1.73h - h = 1.46 \\
  0.73h = 1.46 \\
  h = \dfrac{{1.46}}{{0.73}} \\
  h = 2 \\
 $
Hence, the height of the pedestal is, $ h = 2 $ m.

Note: The figure should be drawn very carefully and the angle should be marked appropriately.
The value of the tangent of the angle should be remembered.