Is the following true or false?From the top and bottom of a building of height h, the angles of elevation of the top of a tower are α and β respectively, then the height of the tower is

htanβ

tanβ−tanα

.

Question

Is the following true or false?From the top and bottom of a building of height h, the angles of elevation of the top of a tower are α and β respectively, then the height of the tower is

htanβ
    
     tanβ−tanα

.

Accepted Answer

Given that,Height of the building=

h
   The angle of elevation of the top of the tower =

α
   and

β
  PQ = the tower RS = the buildingWe draw RT perpendicular to RS

h
   = the height of the building

h'
   = height of the tower,Given that,

α
   = the angle of elevation from the top of the building

∠PRT=α

β
   =the angle of elevation from the bottom of the building

∠PSQ=β
  Then,

PT=QP−TQ

⇒PT=h'−h

Assuming

RT=x
   then

SQ=x
  From the right-angle triangle PRT,

tanα=
      
       PT
      
       RT

⇒tanα=
      
       h'−h
      x

⇒x=
      
       h'−h
      
       tanα
     
     ......(1)

From the right-angled triangle QPS,

tanβ=
      
       PQ
      
       QS

⇒tanβ=
      
       h'
      x

⇒x=
      
       h'
      
       tanβ
     
     ......(2)

From (1) and (2) we have,

⇒
    
     h'−h
    
     tanα
   
   =
    
     h'
    
     tanβ

⇒
    
     h'
    
     tanα
   
   −
    h
    
     tanα
   
   =
    
     h'
    
     tanβ

⇒
    
     h'
    
     tanα
   
   −
    
     h'
    
     tanβ
   
   =
    h
    
     tanα

⇒h'

1
      
       tanα
     
     −
      1
      
       tanβ

=
    h
    
     tanα

⇒h'

tanα−tanβ
        
         tanαtanβ

=
      h
      
       tanα

⇒h'=
      
       htanβ
      
       tanα−tanβ

Thus

h'=
    
     htanβ
    
     tanα−tanβ

is true.Therefore the correct option is 1.

Is the following true or false?

From the top and bottom of a building of height h, the angles of elevation of the top of a tower are α and β respectively, then the height of the tower is $h tan β tan β − tan α$ .

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Is the following true or false?From the top and bottom of a building of height h, the angles of elevation of the top of a tower are α and β respectively, then the height of the tower is htanβ tanβ−tanα .