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Examples I

Example

The angle of elevation of the top of a tower, at a distance 100 m from its foot on a horizontal plane, is found to be 60°. Find the height of the tower.

Solution

Let AB (= h) be the height of the tower.
In the triangle tan 60° = 

        = 
        h = 100 = 173.2 m.
Hence, the height of the tower is 173.2 m.


 

Example

A ladder leaning against a wall makes an angle of 20° with the ground. The foot of the ladder is 3 m away from the wall. Determine the length of the ladder.

 

 

Solution

Let AC (= l) be the length of the ladder in m. In the triangle ABC,  = cos 20°                              

 l = 3 sec 20° = 3[1.064] = 3.192 m
Hence, the length of the ladder is 3.192 m.


 

Example

The shadow of a tower standing on a flat ground is found to be  longer, when the Sun's altitude is 30°, than when it was 60°. Find the height of the tower.

Solution

Let CD = h m be the height of the tower, let x be the length of the shadow when Sun's altitude = 60°.

Length of the shadow =x + 45 when Sun's altitude = 30°

In the ΔBCD,

  =      x =     ....(1)
In the 
Δ ACD,
 = tan 30°

Substituting for x from (1)

    
  
   3h = h + 135
  h = 67.5 m.
Hence, the height of the tower is 67.5 m.

 

 

Example

Two men standing on the opposite sides of a flag staff measure the angles of the top of the flag-staff as 30° and 60°. If the height of the flag staff is 20 m, find the distance between the men.

Solution

Let BC be the flag staff and A and D be the positions of the two men on either sides of the flag staff.

BC = 20 m. 
In the 
Δ ABC, tan 30°= 
          
            AB = 20 m
In the 
Δ DBC, tan 60°= 
            = 
⇒             BD = 
Now  AD = AB + BD
            = 20
            = = 46.19
 The distance between the men is 46.19 m.


 

Example

A flagstaff stands on a top of a 10 m tower. From a point on the ground the angle of elevation of the top of the flagstaff is 60° and that of the tower is 45°. Find the distance of the point of observation (on the ground) from the top of the flagstaff. Also find the height of the flagstaff.

Solution

Let BC be the tower and CD the flagstaff  on top of the tower. From the point of observation A the flagstaff is x m away.

In the Δ ABC,  = tan 45°.
             = 1    

   AB = 10 m.  (1)
In the 
Δ ABD,
                = tan 60°

         

Substituting for AB from (1)
         h + 10 =10 

                h = 10(-1) = 7.32 m
Again , in the  
Δ ABD,
 = cos 60°

         
   x = 20 m.

 

 

Example

Two men are on diametrically opposite sides of a tower. They measure the angles of elevation of the top of the tower as 20° and 24° respectively. If the height of the tower is 40m, find the distance between the two men. 

Solution

In the ΔABD,

          cot 24° = 
          2.246 = 
              AB = 40 x 2.246 = 89.84 m
In the 
Δ CBD,
        cot 20° = 

        2.747 = 
            BC = 40 × 2.747 = 109.88 m
The distance between the two men
                   = AC = AB + BC
                   = 89.84 + 109.88
                   = 199.72 m.

 





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