Hi there.
I've redrawn the diagram and added some extra labels for the sides that will help us. Notice that although it wasn't explicit in your diagram, I have assumed that the rectangularlooking part is indeed a rectangle (all angles are right angles).
The first thing to notice is that the angle formed by sides c and b is congruent to theta. That means the two triangles are similar. Similar triangles have the property that the ratio of corresponding sides is equal. In our diagram, that means z/b = x/c.
From your measurements, we know that b = 36.5  y and from the pythagorean theorem, we know that y = sqrt(x^{2}  z^{2}) and b = 36.5  y. Let's substitute those into the ratios equation:
z/(36.5  sqrt(x^{2}  z^{2})) = x/c
If you put the values z = 118.75 and c = 0.5 into this equation, you get
118.75/(36.5  sqrt(x^{2}  118.75^{2})) = x/0.05, or
36.5x  118.75/2 = x sqrt(x^{2}  118.75^{2})
Squaring both sides and simplifying yields
x^{4} + 15433.8 x^{2}  4334.38 x + 3525.39 = 0
We used Mathematica to solve for x and got four solutions, only one of which makes sense for your situation. The solution is x = 124.09 feet.
Since the cosine of the angle theta is z/x we used the inverse cosine function to find that theta is 0.2945 radians or 16.87 degrees.
Stephen and Penny
