HEIGHT AND DISTANCE math capsule

HEIGHT AND DISTANCE -

Sometimes , we need to find the height of tower , building , tree , distance of a ship from light house, width of river and angle subtended by any object at a given point, etc.

We cannot measure them accurately , though we can find them using the concepts of ,

(1) Angle of elevation
(2) Angle of depression

ANGLE OF ELEVATION -

Angle of elevation is defined as an angle subtended by an eye with the horizontal axis to see an object in the upward direction.


angle of elevation


Let P be the position of the object above the horizontal line OA and O be the eye of the observer, then     ∠AOP is called angle of elevation.

Here, the line joining the eye to the object  i.e. OP is called the line of sight.

IMPORTANT POINTS

* If the angle of observer moves towards the perpendicular line(tower/building) , then the angle of elevation increases and if the observer moves away from the perpendicular line (tower/building) , then the angle of elevation decreases.

* If the angle of elevation of Sun above a tower decreases then the length of shadow of a tower increases.

* If the height of tower is doubled and the distance between the observer and foot of the tower is also doubled, then the angle of elevation remains the same.

Example ,
The string of a kite is 150m long and it makes an angle of 60 degree with the horizontal . The height of the kite from the ground is what ?

Solution,
Let 'h' be the height of the kite from the ground and AB be the length of string = 150m

angle of elevation example


In right angled triangle ABC,

AC =  sin 60
AB

=>  h        =   √3 
      150           2

=>  h = 75 √3m
Hence, the height of kite from the ground is 75 √3m.

ANGLE OF DEPRESSION -

Angle of depression is defined as an angle subtended by an eye with the horizontal axis to see an object in the downward direction.

angle of depression


Let P be the position of the object below the horizontal line OA and O be the eye of the observer, then  ∠AOP is called angle of depression.

In case of angle of depression , the horizontal line through observer must be represented by dotted line.

          * The angle of elevation a point P as seen from a point O is always equal to the angle of depression of O as seen from P.

          * Angle of elevation and depression are always acute angle .

SHORTCUT FORMULAE

1.  If 'b' is height of building and the observer is x units away from base, then,

angle of depression 1


2. If 'DC=b' be the height of a building and Θ1 and Θ2 are elevations measured at B and A, respectively along the same straight line which are x units apart then,

angle of depression 2


Example ,
The angle of depression of two ship from the top of a light house are 45 deg and 30 deg towards East. If the ships are 200m apart , the height of the light house is what?

Solution,
Here applying shortcut method ,

angle of depression example


given, x = 200m , Θ1 = 45 deg,  Θ2 = 30 deg
h =          x           
      cot Θ2 - cot Θ1
putting in the values , we have,

h =   200   m
       √3 - 1 

Here are some more few exercises for you to solve ,


angle of elevation and depression exercise 1


angle of elevation and depression exercise 2

angle of elevation and depression exercise 3

angle of elevation and depression exercise 4

angle of elevation and depression exercise 5

angle of elevation and depression exercise 6

angle of elevation and depression exercise 7

angle of elevation and depression exercise 8

angle of elevation and depression exercise 10

THANKS FOR READING THIS AT Math Capsule 


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