From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 20 m high building are 45 degree and 60 degree respectively. Find the height of the tower.?

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Answer

Answer

Answer

Given:
- The angle of elevation of the bottom of the transmission tower from a point on the ground is 45 degrees.
- The angle of elevation of the top of the transmission tower from the same point on the ground is 60 degrees.
- The transmission tower is fixed at the top of a 20 m high building.

To find:
The height of the transmission tower.

Solution:
Let's assume the height of the transmission tower as 'h' meters.

Step 1: Drawing the diagram:
- Draw a vertical line to represent the building with a height of 20 m.
- From the top of the building, draw a line upwards to represent the transmission tower with a height of 'h' meters.
- From the point on the ground, draw lines to the bottom and top of the transmission tower, forming angles of 45 degrees and 60 degrees, respectively.

Step 2: Identifying the right triangle:
- The vertical line from the top of the building to the top of the transmission tower forms a right triangle with the base being the horizontal distance from the point on the ground to the building.

Step 3: Finding the height of the transmission tower:
We can use the tangent function to find the height of the transmission tower.

In the right triangle formed by the vertical line, the base, and the hypotenuse, we have:
- Tan(60°) = h / x (where x is the horizontal distance from the point on the ground to the building)
- Tan(45°) = (h + 20) / x

Step 4: Solving the equations:
- From the first equation, we can express x in terms of h: x = h / tan(60°)
- Substituting this value of x into the second equation, we get:
Tan(45°) = (h + 20) / (h / tan(60°))
Simplifying, we have: tan(45°) = (h + 20) / (h / √3)
√3 = (h + 20) / h
√3h = h + 20
√3h - h = 20
(√3 - 1)h = 20
h = 20 / (√3 - 1)
Rationalizing the denominator, we can simplify further:
h = 20 * (√3 + 1) / (3 - 1)
h = 10 * (√3 + 1)

Step 5: Final answer:
The height of the transmission tower, h = 10 * (√3 + 1) meters.

Conclusion:
The height of the transmission tower is 10 * (√3 + 1) meters.

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