Height and Distance

Some times we are required to find the height of a tower, tree, building and distance of a ship from light house, width of a river etc. We cannot measure them accurately, though we can find them using the knowledge of trigonometric ratio.

Line of sight: -

When we see an object standing on the ground. The line of sight is the line from our eye to the object, we see.

Angle of Elevation:-

When the object is above the horizontal level of our eye, we have to turn our head upwards to see an object. In this process, our eyes move through an angle which is called angle of elevation.

Angle of Depression:-

When the object is on the ground and the observer is on a building then the object is below the level of the eye of the observer. The observer has to turn his head downward to see the object. In doing so, his eyes move through an angle which is called angle of depression.

Example 1. A man is standing on the deck of a ship, which is 8m above water level. He observes the angle of elevation of the top of a hill as 600 and angle of depression of the base of the hill as 300. Calculate the distance of the hill from the ship and the height of the hill.

Solution: - Let B be man, D the base of the hill, x be the distance of hill from the ship and h + 8 be the height of the hill.

tan 600 = AC/BC

tan 300 = CD/BC

Height of the hill = h + 8 = 24 + 8 = 32m

Distance of the hill from the ship =

Example 2. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 600. When he moves 40m away from the bank, he finds the angle of elevation to be 300. Find the height of the tree and the width of the river.

Solution:- Let height of the tree be y and width of the river be x. CD = 40m

tan 300 = AB/BD

tan 600 =AB/BC

Putting value of y from (ii) to (i)

Height of the tree =

Width of the river = 20m.

Example 3. From a window (h metres high above the ground) of a house in a street, the angle of elevation and depression. of the top and the foot of another house on the opposite side of the street are respectively. Show that the height of the opposite house is .

Solution :- Let W be the window AB the house and WP the width of the street.

Height of the =

Example 4. From the top of a tower 100 m high, the angles of depression of the top and bottom of a pole standing on the same plane as the tower are observed to be 300 and 450 respectively. Find the height of the pole.

Solution:- Let AB be the tower and CD be the pole.

Let CD = h. AB = 100m

Height of the pole = 42.26

Example 5. The angles of elevation and depression of the top and bottom of a light house from the top of a building, 60m high, are 300 and 600 respectively. Find

(i) The difference between the heights of the light house and building.
(ii) Distance between the light house and the building.

Solution:- Let AB be the building and CE be the light house.

AD = BC =

tan 300 = DE/AD

(i) CE - CD = DE = 20m

(ii) Distance = AD = BC = 34.64m

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