Question

If the angle of elevation of a cloud from the point 200 m above the lake is $30^o$ and its angle of depression of its reflection in the lake is $60^o$, then the height of the tower above the lake is-
A. 200 m
B. 500 m
C. 30 m
D. 400 m

Answer 1

Hint: The angle of elevation is the angle above the eye level of the observer towards a given point. The angle of depression is the angle below the eye level of the observer towards a given point. The tangent function is the ratio of the opposite side and the adjacent side.

Complete step-by-step answer:

Let A be the position of the cloud above the lake, which is represented by a dotted line. Let C be the point 200 m above the lake from where the observation is made, and h be the height of the cloud with respect to point C. Also, let BC = x m

By the law of reflection, the object and the image are at equal distances in the opposite sides of the mirror. Hence, this distance is h + 200 m.

We will now apply trigonometric formulas in triangles ABC and DBC, to find the values of h and x.

$\begin{align}
 & In\;\vartriangle ABC, \\
  &\tan {30^{\text{o}}} = \dfrac{{AB}}{{BC}} \\
  &\dfrac{1}{{\sqrt 3 }} = \dfrac{{\text{h}}}{{\text{x}}} \\
  &{\text{x}} = \sqrt 3 {\text{h}}...{\text{(1)}} \\
\end{align} $

$\begin{align}
  &In\;\vartriangle DBC,\; \\
  &\tan {60^{\text{o}}} = \dfrac{{DB}}{{BC}} \\
  &\sqrt 3 = \dfrac{{{\text{h}} + 400}}{{\text{x}}} \\
  &\sqrt 3 {\text{x}} = {\text{h}} + 400...\left( 2 \right) \\
\end{align} $

Substituting (1) in equation (2),
$3h = h + 400$
$2h = 400$
$h = 200 m$

This is the height of the cloud from point C. The height of the cloud from the level of the lake is $200 + 200 = 400 m$

The correct option is D. 400 m.

Note: In such types of questions, it is important to read the language of the question carefully and draw the diagram step by step correctly. When the diagram is drawn, we just have to apply basic trigonometry to find the required answer.