Q vertical pole and a tower are on the same level Ground.from the top of the pole angle of elevation of the top of the tower is 60 degree and the angle of depression od foot of the tower is 30 degree.find the height of the tower if the height of the pole is 20 meter
as usual, draw a diagram. The tower's height is
20 + 20 cot30° * tan60°
= 20(1+tan^2 60°)
= 20 sec^2 60°
= 20 * 4
= 80
To find the height of the tower, we can create a right-angled triangle using the pole, the tower, and the ground. Here are the steps to solve the problem:
Step 1: Draw the diagram:
- Draw a vertical line representing the pole on the ground.
- Draw a line connecting the top of the pole to the top of the tower.
- Draw a horizontal line connecting the foot of the tower to the line representing the ground.
Step 2: Label the known values:
- The height of the pole is given as 20 meters.
- The angle of elevation from the top of the pole to the top of the tower is 60 degrees.
- The angle of depression from the foot of the tower to the top of the pole is 30 degrees.
Step 3: Identify the angles and sides of the labeled triangle:
- Angle A: The angle of elevation (60 degrees) at the top of the pole.
- Angle B: The angle of depression (30 degrees) at the foot of the tower.
- Side AB: The height of the pole (20 meters).
- Side BC: The height of the tower (unknown value that we need to find).
Step 4: Use trigonometric ratios to solve for the height of the tower:
- We can use the tangent function to find BC (the height of the tower).
- tan(A) = AB/BC
- tan(60°) = 20/BC
- √3 = 20/BC
- BC = 20/√3
Step 5: Simplify the expression for BC to get the final answer:
- rationalize the denominator by multiplying both the numerator and denominator by √3:
- BC = 20/√3 * √3/√3 = 20√3/3
Therefore, the height of the tower is approximately 20√3/3 meters.