A Civil Engineer desires to measure the height of a tower

[Stein]

Application of Double Angles (Addition Formulas): A Civil Engineer desires to measure the height of a tower observing a vertical angle at A. He then moves 54 m further towards the tower at B and observes another vertical angle which is twice that of first. He again proceed to move further towards the tower, still on the same line at distance of 21 m. The observed vertical angle of the tower at this point is three times that of the first. Determine the height of the tower
Answer: 33.39 m

Please!!! I need a detailed solution to this problem. Thanks in advance.

 

[Deleted member 4993]

A Civil Engineer desires to measure the height of a tower observing a vertical angle at A. He then moves 54 m further towards the tower at B and observes another vertical angle which is twice that of first. He again proceed to move further towards the tower, still on the same line at distance of 21 m. The observed vertical angle of the tower at this point is three times that of the first. Determine the height of the tower
Answer: 33.39 m

Please!!! I need a detailed solution to this problem. Thanks in advance.

You have posted five problems without showing a line of work!

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[Stein]

To start off.
I'm not sure if its correct. Please try to understand the drawing.

As you can see, I can find the height if I equate the 3 equations. But, I don't know how to equate or what to find first. Please help me.

 

[Deleted member 4993]

To start off.
I'm not sure if its correct. Please try to understand the drawing.
View attachment 4407
As you can see, I can find the height if I equate the 3 equations. But, I don't know how to equate or what to find first. Please help me.

equations to use:

tan(2Θ) = [2tan(Θ)]/[1-tan²(Θ)]

and

tan(3Θ) = tan(Θ+2Θ) → ???

 

[Stein]

tan (Θ+2Θ) = tanΘ+tan 2Θ
--------------
1-tanΘ(tan2Θ)

if i substitute the value of tan 2Θ = 2tan Θ
--------------
1-[(tan)^2 Θ]

i solve it and i arrive in this....

tan (Θ+2Θ) = 3tanΘ-(tan)^3 Θ
----------------
1-3(tan)^2 Θ

am i correct? what will i do then mr.khan?...

 

[Stein]

Mr. Khan??? Can I still have your time? Am I doing fine with this problem if yes then what will i do next? If not, please help me!!!