Trigonometry Problem - Distance between surveyors Sally and Marko are two surveyors that have become separated out in the wilderness. Sally is due east of Marko. Marko radios Sally "The distance from me to the top of Kitt's Peak is 7.8km. It is at an angle of elevation of 32 degrees". Sally radios back "I am 6.5km from the top of the peak". Calculate all possible distances Marko must hike due east in order to reach Sally. Round to the nearest tenth of a kilometer.
I have got an answer of 1.6km by creating a right triangle and using SOH-CAH-TOA but I think I'm wrong because I believe the Law of Sine has to be used, but I do not know how to apply it to this question. Any help is appreciated!
MY WORK: https://imgur.com/a/lfC1frF
 A: My guess is that the person who wrote this question meant for you to assume that Sally and Marko are at the same elevation above sea level, that is, at the same altitude above sea level.
This is only a guess, not a fact stated in the problem, because the term "due east" gives only part of the direction to another point on, above, or below the Earth. The other component of the direction is the angle of elevation. For example, it is possible (although not given) in the statement of the problem that Kitt's Peak is due east of Marko, but we know Kitt's Peak is not at the same elevation above sea level as Marko.
So this is a poorly worded problem that invites us to play a guess-what-the-test-maker-was-thinking game.
I make my guess because without such an assumption, all you really know is that Sally is somewhere in three-dimensional space $6.5$ km from a point that is $7.8$ km from Marko. (You also know the slope of the line from Marko to that point, but since Sally could in principle be looking down onto Kitt's Peak from some higher peak, the slope of one line does not help you at all.)
If you assume that Sally and Marko are at the same elevation above sea level,
then you can work out the sides of a right triangle whose hypotenuse is the line of sight from Marko to Kitt's Peak; one leg is a horizontal line segment of length $7.8 \cos(32^\circ)$ and the other leg is a vertical line segment of length $7.8 \sin(32^\circ),$ as you worked out.
Since Sally is at the same elevation as Marko, the right triangle consisting of one horizontal leg, one vertical leg, and a hypotenuse from her position to Kitt's Peak has the same height as Marko's triangle.
You can use the Pythagorean Theorem to find the length of the horizontal leg.
You also know that the horizontal legs of the two right triangles meet at a point directly underneath Kitt's Peak. Those two legs, and the horizontal line between Marko and Sally, form a third triangle, not necessarily a right triangle (possibly even a degenerate triangle).
You will have worked out two sides of that triangle using the methods shown in your working of the question.
Since the question says nothing about the bearing from anyone to Kitt's Peak,
all you can conclude about the third side is that it must make a triangle of some kind with the other two sides.
That gives you a range of possible answers.

There is still another gaping hole in the presumed logic of the question, which is the assumption that Marko can hike to Sally's position along a perfectly straight line. That's an obviously unrealistic assumption.
So we may have a lower bound on the distance but not an upper bound.
But again we play guess-what-they-meant and conclude that the answer is supposed to assume travel in a straight line.
