What is the height of the tower given following information?

Question

Point C is due East of B and 300 m distance apart. A tower not in line in B and C was observed at B and C having vertical angles of 45 deg and 60 deg respectively. The same tower was observed at point D 500 m west of B. The vertical angle of the same tower as observed from D is 30 deg. The height of the tower is.

Below is my initial figure and equation.

tan 30 = H/LD; tan 45 = H/LB; tan 60 = H/LC

After that, i'm not sure how to proceed.

Answer 1

Is this a three-dimensional geometry problem? Can we assume the tower is perpendicular to the ground and the ground is a flat plane with no changes in elevation?

If so, then we have $$H = L_C \tan 60^\circ = L_B \tan 45^\circ = L_D \tan 30^\circ,$$ as you wrote, which upon evaluation of the tangent functions yields $$H = L_C \sqrt{3} = L_B = \frac{L_D}{\sqrt{3}}.$$ Squaring yields $$H^2 = 3 L_C^2 = L_B^2 = \frac{L_D^2}{3}.$$ Now we need to find a relationship between the three distances to the tower that does not depend on $H$. By Stewart's theorem, we have $$500 L_C^2 + 300 L_D^2 = 800(L_B^2 + 150000),$$ hence $$5 L_C^2 + 27 L_C^2 = 8(3 L_C^2 + 150000),$$ or $$L_C = 100 \sqrt{15}$$ and $$H = L_C \sqrt{3} = 300 \sqrt{5}.$$

Answer 2

Let the feet of the tower be $P$ and let it's height be $h$.

Now, express $PC$, $PD$ and $PB$ in terms of $h$ and then apply Stewart's Theorem on $\triangle PCD$ and solve for $h$.