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Q.

From a point on the level ground, the angle of elevation of the top of a tower is 30 °  . On proceeding 30 m towards the tower the angle of elevation becomes 60 °  . What will be the height of the tower?


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a

5 3    m

b

15 2    m

c

15 3    m

d

10 3    m 

answer is C.

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Detailed Solution

Given,
Angle of elevation (θ)= 30 °   Distance moved towards the tower y=30   m
The new angle of elevation from the ground (ϕ)= 60 °   We draw a schematic diagram and get a relation connecting the height of the tower and the angle of elevation θ   .
Question ImageIn ΔPQS  ,
PQ be the tower and QS be the ground.
x   be the distance from the bottom of the tower to the position at which the angle of elevation of the top of the tower is 60 °  .
In ΔPQS  we use tanθ= height base  ,
tanθ= PQ QS ⇒tanθ= PQ QR+RS ⇒tanθ= h x+30  
Substituting θ  = 30 °   in tanθ= h x+30  ,
⇒tan 30 ° = h x+30    ⇒ 1 3 = h x+30    [∵tan 30 ° = 1 3 ] ⇒h= x+30 3  
Considering ΔPQR   and get another relation connecting the angle of elevation ϕ   and the height of the tower,
tanϕ= PQ QR ⇒tanϕ= h x  
Substituting ϕ  = 60 °   in tanϕ= h x   ,
⇒tan 60 ° = h x ⇒ 3 = h x      [∵tan 60 ° = 3 ] ⇒h= 3 x  
Equating the relations h= x+20 3   and h= 3 x  ,
⇒ x+30 3 = 3 x ⇒x+30= 3 × 3 x ⇒x+30=3x ⇒30=3x−x   ⇒30=2x ⇒ 30 2 =x ⇒x=15  
Substituting x=15 m   in  h= 3 x  ,
⇒h= 3 ×15 ⇒h=15 3  m  
Thus the height of the tower will be 15 3    m.
Therefore the correct option is 3.
 
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