Question

At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is 5/12. On walking 192 metres towards the tower, the tangent of the angle of elevation is 3/4. Find the height of the tower.and find next answer also​

Answer 1

Given :-

• At a point (say, C) the tangent of the angle of elevation of the vertical tower is 5/12.
• At another point (say, D) which is 192 metres away from C, the tangent of the angle of elevation of the same tower is 3/4.

To Find :-

• The height of the tower.

Solution :-

Figure:-

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Let AB be the tower and let the angle of elevation of its top at C be ɑ. Let D be a point at a distance of 192 metres from C such that the angle of elevation of the top of the tower at D be β. Let h be the height of the tower and AD = x.

$\hrulefill$

We are given that,

⇒ tan ɑ = 5/12

⇒ tan β = 3/4

In △ CAB, we have,

$\implies\rm{tan\:\alpha=\dfrac{AB}{AC}}$

$\implies\rm{tan\:\alpha=\dfrac{5}{12}=\dfrac{h}{x+192}\longrightarrow{(1)}}$

In △ DAB, we have,

$\implies\rm{tan\:\beta=\dfrac{AB}{AD}}$

$\implies\rm{tan\:\beta=\dfrac{3}{4}=\dfrac{h}{x}\longrightarrow{(2)}}$

From equation ( 2 ), we have,

$\implies\rm{3x=4h}$

$\implies\rm{x=\dfrac{4h}{3}}$

Substituting value of x in equation ( 1 ), we get,

$\implies\rm{\dfrac{5}{12}=\dfrac{h}{192+4h/3}}$

$\implies\rm{5\left(192+\dfrac{4h}{3}\right)=12h}$

$\implies\rm{5(576+4h)=36h}$

$\implies\rm{2880+20h=36h}$

$\implies\rm{16h=2880}$

$\implies\rm{h=\dfrac{2880}{16}=}\:\bold{180.}$

Hence, the height of the tower is 180 metres.