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The angle of elevation of the top of a tower from a point A on the ground is ${30^ \circ }$. On moving a distance of $20$ metres towards the foot of the tower to a point B the angle of elevation increases to ${60^ \circ }$. Find the height of the tower and the distance of the tower from point A.

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Hint : (Make figure , analyze from it , get the problem solved)


From the figure
AB$ = 20m$ (Given)
Given
$
  \angle CAB = {30^ \circ } \\
  \angle CBD = {60^ \circ } \\
 $
To find CD, The height of the tower
First we will consider triangle ACD
We know ,
$
  \tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }} = \dfrac{{CD}}{{AD}} \\
  AD = \sqrt 3 \,CD\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......({\text{i}}) \\
 $
In triangle CBD we have
$
  \tan {60^ \circ } = \dfrac{{CD}}{{BD}} = \sqrt 3 \\
  \dfrac{{CD}}{{\sqrt 3 }} = \,BD\, \\
    \\
  AB = AD - BD \\
  20 = \sqrt 3 \,CD - \,\,\dfrac{{CD}}{{\sqrt 3 }} = CD\,\left( {\dfrac{2}{{\sqrt 3 }}} \right)\, \\
  \therefore \,\,CD = 10\sqrt 3 \\
 $
Hence the height of the tower is $10\sqrt 3 = 17.34\,\,m$
Now we have to find the Length AD
So, Again we can consider the triangle ADC
$
  tan{30^ \circ } = \dfrac{1}{{\sqrt 3 }} = \dfrac{{10\sqrt 3 }}{{AD}} \\
  AD = 10\sqrt 3 .\sqrt 3 = 30\,\,m \\
$
Hence the distance of the tower from point A is 30 m.

Note :- Whenever these types of questions arise you must have to use the trigonometric concepts for triangles . Using the values of respective angles you can simply find any length present in the figure using “tan” in most of the cases will make your problem solved easily .