with only the 200 meters height and 5º depression angle, how to discover the curve D (distance from the lighthouse to the boat, on earths surface)

Question

with only the 200 meters lighthouse and 5º depression angle of view, how to discover the curve D, distance from the lighthouse to the boat, on earths surface, having in consideration the earths curvature due to the long distances visualization

Answer 1

Using the sine and cosine rules on a triangle with vertices on the center of the earth $O$, the top of the lighthouse $L$ and the boat $B$, and labelling $L\hat{O}B=\theta, O\hat LB=w, LB=\gamma$ and $R,h$ are set to be the radius of the Earth and the height of the lighthouse respectively, we obtain two equations for two unknowns:

$$\gamma^2=R^2+(R+h)^2-2R(R+h)\cos\theta$$$$\frac{\sin w}{R}=\frac{\sin\theta}{\gamma}$$

Our goal is to calculate $\theta$, since the distance required is given by $D=R\theta$. One can find an expression by solving the second equation for $\gamma$ and substituting into the first equation. This results in a quadratic equation for $\cos\theta$ with the solution

$$\cos\theta=\left(1+\frac{h}{R}\right)\sin^2w+\cos w\sqrt{1-\left(1+\frac{h}{R}\right)^2\sin^2w}$$

In your case, we readily see that $h=200~\text m~, w=85^\circ$. We don't need the radius of the Earth to produce a reasonable approximation to this however. Assuming that $\theta$ is small enough so we can use the Taylor expansion of the LHS and Taylor expanding the RHS in a series of $h/R$ we obtain the expression

$$D\approx\frac{h\tan w}{\sqrt{2}}\approx 1612 ~\text m$$

for the distance of the boat.