I have a problem with a trig question about distance between hikers.

[ErionK]

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 don't know how to begin the question, I don't understand it at all.

 

[Otis]

Hi Erion. I think we're supposed to assume that Kitt's Peak is located either in-between the surveyors or east of both of them. It doesn't really make sense to ask about Marko hiking due east to meet Sally if she's on the other side of the peak (unless there's a tunnel, heh). Yet, we can still find the horizontal distances possible between them. The diagram shows how I interpret the question, where we would need to calculate the distance from M to S₁ and from M to S₂.

MOT and both SOTs are right triangles. They gave you distances [imath]\overline{MT}[/imath] and [imath]\overline{TS}[/imath]..................edited

Please share your work, if you have any questions.

 

[The Highlander]

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 don't know how to begin the question, I don't understand it at all.

Hi @ErionK,

@Otis' analysis seemed perfectly valid to me at first glance but thinking about it a bit further it occurred to me to ‘visualise' the situation in a different way:-

First, let’s assume you are an observer at ground level and the surveyors are directly opposite each other (it doesn’t matter that they really aren’t). I am going to use the last letter of Sally’s name (because the upper & lower case “s” are so similar).

This can now produce a sketch of what might be called a Front Elevation where:-
P is the top of Kitt’s Peak.
Z is “Ground Zero” the bottom of a line dropped vertically from P to ground level.
M is Marko’s position, Y is Sally’s and
m & y are their respective distances from Z.
(You should reproduce this sketch (or similar) in your work.)

​

Looking at the sketch (above), it should be clear to you how to calculate (using basic Trig.) the distance from P to Z (the vertical height of Kitt’s Peak) and thence (using a well-known ‘theorem’) the distances m & y that the surveyors are away from Ground Zero (Z).

Once you have those distances, you can now sketch a Plan (‘bird's eye view’) looking down on the top of Kitt’s Peak.
Can you now see that two concentric circles (centred at Z) with respective radii of m & y will provide the locus of possible positions for each surveyor in turn?

​

However, this analysis leads me to conclude that there are an infinite number of possible distances between Marko & Sally! (Unless you have omitted a vital piece of information that was contained in the original problem!)

I would think that, for the problem to be “solvable” it must give some information about the position of one of the surveyors to the Peak, eg: “Marko is due South-West of Kitt’s Peak” or “Sally is due North-East” (or some other reference to their positions relative to Z).

Does the original problem not provide this?

 

[Steven G]

Hi Erion. I think we're supposed to assume that Kitt's Peak is located either in-between the surveyors or east of both of them. It doesn't really make sense to ask about Marko hiking due east to meet Sally if she's on the other side of the peak (unless there's a tunnel, heh). Yet, we can still find the horizontal distances possible between them. The diagram shows how I interpret the question, where we would need to calculate the distance from M to S₁ and from M to S₂.

MTO and both STOs are right triangles. They gave you distances [imath]\overline{MT}[/imath] and [imath]\overline{TS}[/imath].

Please share your work, if you have any questions.

View attachment 32648

To OP (and Otis).
Otis meant to write that both MOT and SOT are right angles. ......................................fixed

 

[The Highlander]

To OP (and Otis).
Otis meant to write that both MOT and SOT are right angles.

Yes, I had already read his submission as meaning that MTO and both S₁TO & S₂TO (his "STOs" shorthand) were all right angles.
Unfortunately, that doesn't alter the potential "problem" I see with his analysis; it actually reinforces my concern(s). ??

 

[Dr.Peterson]

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 don't know how to begin the question, I don't understand it at all.

To me, the missing information is that S and M are both on a plain around the mountain, and at the same elevation. I initially pictured then as on the slopes.

Given this, you can find the height of the mountain and the horizontal distance of each from it, as Highlander says. Then, since we have no information other than that, I take the question as asking, not for two possible distances, but for an interval, based on the triangle inequality. (This would include the possibility of having to climb over the mountain.)

Or, of course, there may be other information omitted!

 

[JeffM]

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 don't know how to begin the question, I don't understand it at all.

It is perfectly reasonable for you not to understand the question. It makes absolutlely no sense.

 

[The Highlander]

To me, the missing information is that S and M are both on a plain around the mountain, and at the same elevation. I initially pictured then as on the slopes.

Hello again, Dr.P. ?

I think it’s a perfectly reasonable assumption (indeed, the actual intention of the problem's author, I suspect) that the surveyors are both on “flat” ground well away from the slopes of the mountain, especially given the substantial distances they are from the summit (Everest is only c.8.8km high ?).

I really don’t believe that is (if any) the “missing information”.

Applying the KISS principle (Occam's razor, if you prefer), I think it’s just a matter of the OP failing to supply something that will define the angle θ in my revised loci diagram, below (qv).
For example, if we are told that Marko is due SW of the Peak then θ=45° and the distances to the two possible positions (Y₁ & Y₂) for Sally can then easily be calculated.

Whilst I take your point about “an interval, based on the triangle inequality” the question does (ostensibly) ask for: “all possible distances” (plural) and trying to calculate the distance “to climb over the mountain” would involve a further myriad of 'missing information' (unless the surveyors inhabit a world of perfectly straight lines! ?)

​

No, my bet is that the OP has in his/her “summary” of the question omitted something that will define the angle θ.

 

[The Highlander]

It is perfectly reasonable for you not to understand the question. It makes absolutlely no sense.

It makes a little bit of sense, especially if the OP (as is their wont ?) has just failed to provide us with one small piece of extra information. ?