SOLUTION: A surveyor standing W 25 degrees S of a tower measures the angle of elevation of the top of the tower as 40 degrees 30. From a position E 23 degrees S from the tower the l elevatio

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Question 1201107: A surveyor standing W 25 degrees S of a tower measures the angle of elevation of the top of the tower as 40 degrees 30. From a position E 23 degrees S from the tower the l elevation of the top is 37 degrees 15. Determine the height of the tower if the distance between the two observations is 75m.
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

Notation like 40°30' is read out as "40 degrees 30 minutes".
"minutes" refers specifically to arc minutes

There are 60 arc minutes in a full degree.
30 arc minutes is 30/60 = 1/2 = 0.5 of a degree.
15 arc minutes is 15/60 = 1/4 = 0.25 of a degree.

Therefore,
40 degrees 30 minutes = 40.5 degrees
37 degrees 15 minutes = 37.15 degrees
which represent the angles of elevation.

This is what it looks like from a birds eye view

A = location of the surveyor
B = second observation location
C = location of the tower
Notation like W 25° S means "aim directly west, then turn 25° toward the south"
The blue dashed line in the diagram above helps us set up the angles of 25° and 23°

Refer to this page for a few other examples
http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html

Also, refer to this problem involving similar compass bearings notation.
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1200155.html

Angle ACB is sandwiched between the 25 and 23 degree angles.
Those three angles add to 180.
25+(angleACB)+23 = 180
angleACB + 48 = 180
angle ACB = 180-48
angle ACB = 132

Let's update the drawing with that angle.

Here's a 3D look at what's going on.
Pay close attention to the points A,B,C how they are laid out.
Compare their relative locations to the previous diagram shown above.

triangle ABC is along the flat ground
A = location of the surveyor
B = second observation location
C = base of the tower
D = top of the tower
x = height of the tower = length of segment CD
angle ACB = 132 degrees
angle CAD = 40.5 degrees
angle CBD = 37.25 degrees

Here is the side view of right triangle ACD

Here is the side view of right triangle BCD

At this point I get stuck.
I don't think there's enough information to determine x.
If we knew the length of segment AC or the length of segment BC, then we could use trigonometry to determine x.