In this page trigonometry practical problem10 we are going to see solution of first problem with detailed explanation.
Question 10:
The angles of depression of the top and the bottom of a 12 m high building from the top of the tower are 45 degree and 60 degree respectively. Calculate the height of the tower
Solution:
First let us draw the rough diagram for the above question.
here AD represents height of the tower , DC represents the distance between tower and the building.
In the right triangle ABC the side which is opposite to angle is known as opposite side(AB),the side which is opposite to 90 degree is called hypotenuse side(AC) and remaining side is called adjacent side (BC).
In the right triangle ADE the side which is opposite to angle is known as opposite side(AD),the side which is opposite to 90 degree is called hypotenuse side(AE) and remaining side is called adjacent side (DE).
Now
we need to find the height of tower (AD)
tan θ = Opposite side/Adjacent side
Let AB = h
In ∆ ABC
tan 45 = AB/BC
1 = x/BC
BC = x  (1)
In ∆ ADE
tan 60 = AD/DE
√3 = (AB+BD)/DE
√3 = (12 + x)/DE
DE = (12 + x)/√3  (2)
Length of BC and DE are equal
(1) = (2)
x = (12 + x)/√3
√3 x = 12 + x
√3 x  x = 12
x(√3  1) = 12
x = 12/(√3  1)
x = [12/(√3  1)] x [(√3 + 1)/(√3 + 1)]
x = [12(√3 + 1)] / [(√3)²  1²)]
x = [12(√3 + 1)] / 2
x = [6(√3 + 1)]
x = 6(1.732 + 1)
x = 6(2.732)
x = 16.392
Height of the tower = 12 + 16.392
= 28.392 m Trigonometry Practical Problem10 Trigonometry Practical Problem10
Application Problems in Trigonometry

Solution 
(1) The angle of elevation of the top of the building at a distance of 50m from its foot on a horizontal plane is found to be 60 degree. Find the height of the building.  
(2) A ladder placed against a wall such that it reaches the top of the wall of height 6 m and the ladder is inclined at an angle of 60 degree. Find how far the ladder is from the foot of the wall.  
(3) From the top of the tower 30m height a man is observing the base of a tree at an angle of depression measuring 30 degree. Find the distance between the tree and the tower.  
(4) A man wants to determine the height of a light house. He measured the angle at A and found that tan A = 3/4. What was the height of the light house if A was 40m from the base?  
(5) The angle of elevation of the top of the light house at a distance of 30 m from its foot on a horizontal plane is found to be 60 degree. Find the height of the light house.  
(6) A man is standing on the top of a multistoryed building 45 m high is looking at two advertising pillars on the side whose angle of depression are 30 degree and 45 degree. What was the distance between two pillars.  
(7) Two men are on the opposite sides of a building. They measure the angles of elevation of the top of the building as 30 degree and 60 degree respectively. If the height of the building is 150 m, find the distance between the men.  
(8) The angle of elevation of a multistoreyed building from a point on the toad changes from 30 degree to 60 degree as one walks 120 m along the road towards the building,find the height of the building.  
(9) A flag staff stands on the top of 6 m high tower. From a point on the floor the angle of elevation of the top of the flag staff is 60 degree and from the same point the angle of elevation of the top of the tower is 45 degree. Find the height of the flag staff. 