6.The angle of elevation to the tower is 30(degree) and then moved towards the tower of a distance 20m. Now the angle of elevation is 60(degree). What is the height of the tower?

Make a sketch

Label the top of the tower as P and its base Q
Label the first position of observer as A, his second position as B
Join the straight line ABQ
In triangle ABP, angle A = 30
angle ABP = 120
thus angle APB = 30, making AB = BP
In triangle PBQ
sin60 = PQ/30
PQ = 30sin60 = ...

17.32m

To find the height of the tower, we can use trigonometric ratios and basic geometry.

Let's consider a right-angled triangle formed by the observer, the base of the tower, and the top of the tower.

Given:
Angle of elevation from the observer to the tower = 30 degrees
Angle of elevation after moving towards the tower = 60 degrees
Distance moved towards the tower = 20 m

Step 1: Finding the height of the triangle
In the triangle, we have two sides and one angle of elevation. We need to find the height of the triangle.

We can use the tangent function:
tan(angle) = opposite/adjacent

For the angle of elevation of 30 degrees:
tan(30 degrees) = opposite/adjacent
tan(30 degrees) = height/base ...(1)

For the angle of elevation of 60 degrees:
tan(60 degrees) = opposite/adjacent
tan(60 degrees) = height/(base - 20) ...(2)

Step 2: Solving the equations
Now we have a system of two equations with two unknowns (height and base). We can solve these equations simultaneously.

From equation (2), we can rewrite it as:
height = tan(60 degrees) * (base - 20)
height = (√3 / 3) * (base - 20) ...(3)

Substituting the right-hand side of equation (3) into equation (1):
tan(30 degrees) = (√3 / 3) * (base - 20)
√3 / 3 = (√3 / 3) * (base - 20)
base - 20 = 3
base = 23

Step 3: Calculating the height of the tower
Now that we have the base of the triangle (23m), we can substitute it into equation (1) to find the height:

height = tan(30 degrees) * base
height = (√3 / 3) * 23
height ≈ 13.34m

Therefore, the height of the tower is approximately 13.34 meters.