10 Math Height&distance

Finish Line & Beyond HEIGHT & DISTANCE Line of Sight: The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. Angle of Elevation: The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object Angle of Depression: The angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed. Height or Length: The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios. Example: A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower. Solution :

A Angle of Depression

Angle of Elevation

B

C

In the given figure AB is the height of tower, BC is the distance between base of tower and the viewer and ∠ C is the Angle of Elevation. By using tan C the height of the tower can be calculated. BC =15 m ∠ C = 60° AB = ?

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Finish Line & Beyond Now, tan 60° =

AB BC

AB 15 Or, AB = 15 3 meter is the answer Or,

3 =

Example: An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3m below the top of the pole to undertake the repair work. What should be the length of the ladder that she should use which, when inclined at an angle of 60° to the horizontal, would enable her to reach the required position? Also, how far from the foot of the pole should she place the foot of the ladder? (You may take 3 = 1.73) Solution : This can be solved using the same figure as given in example A, with some changes.

A D

60° B

C

AB = 5 m ∠ C = 60° DC=? Is the length of the ladder. BC=? Let us assume the electrician needs to reach up to a point D on line AB, so as per the question, BD= 5-1.3=3.7 m

AD DC 3.7 3 Or, = DC 2 3.7 × 2 Or, DC = = 5.69 m, length of the ladder 1.3 sin 60° =

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Finish Line & Beyond AD BC 3.7 Or, 3 = BC Or, BC =3.7 ÷ 1.3 = 2.84 meter is the distance of the base of ladder and that of pole. tan 60° =

Example: An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney? Solution: A

C

B

D

E

In the given figure AD is the chimney, CE is the height of the observer and DE or BC is the distance of the chimney from observer. As BD= CE, so AB+CE will be equal to the height of the chimney.

AB BC AB Or, 1 = 28.5 tan 45°=

Or, AB = 28.5 So, AD = 28.5+1.5 = 30 meter is the height of the chimney

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