Mind Map

Introduction:
   In our daily life, some times we need to measure the height of a building, tree, etc.
   How were these heights measured? can you measure the height of your house?
   Let us understand through some examples, Shiva wants to find the height of an electrical pole. He tries to locate the top most point of the pole. He also imagines a line joining the top most point and his eye.
   This line is called 'line of sight'. He also imagines a horizontal line, parallel to earth, from his eye to pole. 
   Here 'the line of sight', 'horizontal line' and 'the pole' form a right angle triangle. The line of sight is above the horizontal line and angle between the line of sight and the horizontal line is called 'angle of elevation'.
   Suppose you are standing on the top of your school building and you want to find the distance of borewell from the building on which you are standing. For that, you have to observe the base of the borewell.
   Then, the line of sight from your eye to the base of borewell is below the horizontal line from your eye.
   Here the angle between the line of sight and horizontal line is called 'angle of depression'.
 

Drawing Figures for Given Situation:
   When we want to solve problems of heights and distances, we should consider the following.
* All objects such as towers, trees, buildings, ships, mountains, etc., Shall be considered as linear for mathematical convenience.
* The angle of elevation or angle of depression is considered with reference to the horizontal line.
* The height of the observer is neglected, if it is not given in the problem.
   To find heights and distances, we need to draw figures and with the help of these figures we can solve the problems.
 

Problems

1. The top of a clock tower is observed at an angle of elevation θ° and the foot of the tower is at the distance of 'd' meters from the observer. Draw the diagram for this data.

2. Lakshmi observes a flower on the ground from the balcony of the first floor of a building at an angle of depression α°. The height of the first floor of the building is 'h' meters. Draw the diagram for this data.

3. A balloon has been tied with a rope and it is floating in the air. A person observed the balloon from the top of a building at angle of elevation θ₁ and foot of the rope at an angle of depression θ₂. The height of the building is 'h' feet. Draw the diagram for this data.

4. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 30°. Find the height of the tower.
Sol: Let AB be the tower and C be a point on ground such that the angle of elevation of the top A of tower AB is 30°.
Here height of the tower AB = h m (say)
Distance between observer and foot of the tower BC = 30 m
Angle of elevation at C is θ = 30°

5. Ramu observes a boy standing on the ground from a helicopter at an angle of depression 45°. If the helicopter flies at a height of 50 meters from the ground, what is the distance of the boy from Ramu?

6. An observer of height 1.8 m is 13.2 m away from a palm tree. The angle of elevation of the top of the tree from his eye is 45°. What is the height of the palm tree?

Angle of elevation at D is 45°

Height of the palm tree AB = AE + BE
                                                = 13.2 + 1.8 = 15 m
 

7. A water tank is placed on top of a tower. From a point on the ground 40 m away from the foot of the tower, the angle of elevation of the top of the tower is 30°. The angle of elevation of the top of the water tank is 45°. Find the (i) height of the tower (ii) the depth of the tank.
Sol: Let BC is the tower of height = h1 m
CD be the height of water tank = h2 m
A be a point on ground at a distance of 40 m away from foot B of the tower.

⇒ 40 = 23.09 + h₂ 
∴ h₂ = 40 − 23.09 = 16.91 m
∴ Height of the tower BC = 23.09 m and depth of the tank CD = 16.91 m

Writer: T.S.V.S. Suryanarayana Murthy