Question

The height of a tower is 100 m. When the angle of elevation of the sun changes from $30^o$ to $45^o$, the length of the shadow becomes x metres less. The value of x is-
$
  {\text{A}}.\;100\;{\text{m}} \\
  {\text{B}}.\;100\sqrt 3 \;{\text{m}}\; \\
  {\text{C}}.\;100\left( {\sqrt 3 - 1} \right)\;{\text{m}} \\
  {\text{D}}.\dfrac{{100}}{{\sqrt 3 }}{\text{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:

The sun moves from $30^o$ to $45^o$ along the dotted line. The shadow moves from point F to E, covering a distance x m as given in the question. CD is the tower of height 100 m. We will apply trigonometric formulas in triangles DCE and DCF to find the value of x.

$

  In\;\vartriangle DCE, \\

  \;\tan {45^{\text{o}}} = \dfrac{{CD}}{{CE}} \\

  1 = \dfrac{{100}}{{CE}} \\

  CE = 100\;{\text{m}} \\

$
$
  In\vartriangle DCF, \\

  \;\tan {30^{\text{o}}} = \dfrac{{DC}}{{CF}} \\

  \dfrac{1}{{\sqrt 3 }} = \dfrac{{DC}}{{CE + EF}} \\

  Substituting\;the\;value\;of\;CE, \\

  \dfrac{1}{{\sqrt 3 }} = \dfrac{{100}}{{100 + {\text{x}}}} \\

  100 + {\text{x}} = 100\sqrt 3 \\

  {\text{x}} = 100\sqrt 3 - 100 \\

  {\text{x}} = 100\left( {\sqrt 3 - 1} \right)\;{\text{m}} \\

$
This is the value of x. The correct option is C.


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.