A question on the solution for the lifeguard problem (or Snell's law)

Question

This question is about the lifeguard problem as presented on the linked assignment, essentially this is related to Snell's law. In the solution, after introducing appropriate names, the condition $$ \sin \theta_1 = n \sin \theta_2 $$ is derived. But how to get the coordinate $x$ which was asked for in the initial problem from this condition? And is this unique, or are there other rays (maybe having part of it outside the pictured region) with these angles such that the above relation holds? The above relation has two parameters, but in the original problem we are just looking for one parameter, so I am not sure that there is a 1-1 relation between both.

Answer 1

Snell's law was derived from a basic proposition that that light follows path of least time. (Fermats principle)

The Snell's law is:

$$\dfrac{c}{v_1}\sin(\theta_1)=\dfrac{c}{v_2}\sin(\theta_2)$$

where $c, v_1, v_2$ are speeds of light in vaccuum, medium $1$ and medium $2$ respectively.

Thus in your problem, we may say,

$$\dfrac{\sin(\theta_1)}{v_r}=\dfrac{\sin(\theta_2)}{v_s}$$

Now you haven't been provided with either of the angles, but only the angle that straight line joining the two people makes with normal.

It turns out that this information is not enough for solving the problem explicitly.