Trigonometry Problem Solving How can we estimate the height (h) of a castle surrounded by a moat, using the info below?

 A: From the diagram, $tan30^\circ$ = $\frac {h}{x+45} $ where x denotes the distance (in meters) from the closer viewing point to the wall.
also,
$tan50^\circ$ = $\frac {h}{x} $
we get,
$x+45=\frac{h}{\tan 30^\circ}$
$x=\frac{h}{\tan 50^\circ}.$
subtracting the equations, we get,
$45= h\left(\frac1{\tan30^\circ}-\frac1{\tan50^\circ}\right).$
You're done!!
A: I assume that's supposed to be $50^\circ,$ not just $50$. Let $x$ be the distance (in meters) from the closer viewing point to the castle wall, so that the distance to the further viewing point is $x+45$ meters. Use SOHCAHTOA on the two right triangles and solve the system of two equations in $x$ and $h$ for the variable $h$. (Don't estimate until the very end.)
A: I have another way to solve this which is completely different but still gives the correct answer. Its just the first method that immediately came to my mind when I saw the question.
I look at the right triangle as two separate triangles. First I used law of sines on the triangle to the left to find the hypotenuse of the whole right triangle, the distance from the 30Deg measurement to the top of the tower. 
We have to find all the angles. Of course they all have to add up to 180 degrees:
$180-50=130$
$130+30=160$
$180-160=20$
https://pp.vk.me/c604326/v604326834/130f9/VZT0aNwcuuo.jpg
I now have all the angles for the triangle on the left to find the hypotenuse. Now we use law of sines. 
hypotenuse = $45m(\sin(130\deg)/sin(30\deg))= 100.7893$
To find the proportion of the adjacent to the hypotenuse (which we already know and can just multiply the value), we use cosine. 
$\cos(30\deg)*100.7893 = 87.286$
Then we use tangent to find that distance in proportion to the opposite (the height of the tower)
$\tan(30\deg)=0.57735026919$
$87.286*0.57735026919= 50.39$ meters
Then we can use pythagoras theorem to check that our answer is correct:
$\sqrt{87.286^2+50.39^2}=100.78$ meters
This gives me the same exact answer for the previously solved
$45/(1/\tan(30\deg)-1/\tan(50\deg)))$
