Simone is facing north and facing the entrance to a tunnel through a mountain. She notices that a $1515$ m high mountain is at a bearing of $270^\circ$ from where she is standing and its peak has an angle of elevation of $35^\circ$. When she exits the tunnel, the same mountain has a bearing of $258^\circ$ and its peak has an angle of elevation of $31^\circ$.

Assuming the tunnel is perfectly level and straight, how long is it?

3d diagram of problem

I had two problems with this question. I was getting two different answers using methods that should give the same answer and neither of those answers matched with the answer in the textbook.

Attempt 1:

What we want to figure out is the value for $d$. If we can figure out the values for $x$ and $y$, then we can use Pythagorean theorem to figure out the value for $d$.

In this case,

$$x = \frac{1515}{\tan 35^\circ} \qquad\text{and}\qquad y = \frac{1515}{tan 31^\circ}$$

We also know that $d =\sqrt{y^{2}-x^{2}}$, so

$$d = \sqrt{\left(\frac{1515}{\tan 31^\circ}\right)^{2}- \left(\frac{1515}{\tan 35^\circ}\right)^{2}}$$

This makes the value of $d$ about 1294 meters.

Attempt 2:

We can figure out the value for $\theta$. In the ground level triangle $\theta = 258^\circ -180^\circ = 78^\circ$. This also means $\gamma = 90^\circ - 78^\circ = 12^\circ$. In the solution above, we figured out the value for $x$ and $y$. We can use trig ratios to figure out the value for $d$.

$$\tan\gamma = \frac{d}{x} \qquad\to\qquad d = x \tan 12^\circ = \frac{1515 \tan 12^\circ}{\tan 35^\circ}$$

This gives a value for $d$ equal to about $460$ meter.

In my textbook, the answer for the length of the tunnel is actually $650$ meters. I was wondering what am I doing wrong. Also: Why are my two answers not matching?

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    $\begingroup$ Do you understand that, since the angle given is 270 degrees, the triangle "on the ground" is also a right triangle? $\endgroup$ – user247327 May 20 '16 at 0:09
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    $\begingroup$ Yes I know that. Thats why i used trigonometric ratios in attempt 2. $\endgroup$ – user262291 May 20 '16 at 0:09
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    $\begingroup$ The numbers in the question are contradictory. From just 1515, 35, 31 and 270 they fix all the other lengths and angles $\endgroup$ – M.M May 20 '16 at 0:51
  • $\begingroup$ That is pretty strange. I got this question from a textbook so you would expect that they would look over whether there numbers make sense. $\endgroup$ – user262291 May 20 '16 at 0:53
  • $\begingroup$ It is the simplest explanation, though. If the input is inconsistent then of course you'll arrive at different answers by different methods. The only way I can see to squeeze an extra variable out of the problem is to interpret "facing the entrance" to mean only that her position is at the entrance, but she is facing it obliquely, so the tunnel itself has some angle away from due north. At first I thought maybe the tunnel was not at elevation 0, but this turns out to make much of a difference. $\endgroup$ – Erick Wong May 20 '16 at 0:57

Notice that in your first solution you do not need the asserted angle of $258^\circ$.

In fact, from the information given you can conclude that the angle must actually be $239.1^\circ$.

So those who said the information was inconsistent were correct. The $258^\circ$ is bogus.

You should have been asked to find the angle since enough information is given to find it.

But it is $239.1^\circ$ not $258^\circ$ as claimed in the problem.

Diagram of tunnel problem

  • $\begingroup$ This still does not give d=650 however $\endgroup$ – M.M May 20 '16 at 12:39
  • $\begingroup$ @M.M Nor should it. The values $258^\circ$ and $31^\circ$ are mutually contradictory. Whichever one you use makes the other bogus. $\endgroup$ – John Wayland Bales May 20 '16 at 13:06
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    $\begingroup$ My point is that you need to replace both of them to be able to get the "correct" answer of 650, so this problem is more broken than just one bad number. $\endgroup$ – M.M May 20 '16 at 13:22
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    $\begingroup$ If you take the book's incorrect answer of 650 as "given" and the problem as wrong, then you can work backwards from 650 to correct the both wrong angles stated in the problem. You would have to find the "correct" value of $258^\circ$ and the corresponding "correct" value of $31^\circ$. The values of the two angles are not independent. Or, you could take the $31^\circ$ to be correct and the $35^\circ$ to be incorrect and try to find its "correct" value. Do you see how pointless this approach is and that the only reasonable conclusion is that both the problem and the alleged answer are in error $\endgroup$ – John Wayland Bales May 20 '16 at 13:29
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    $\begingroup$ One more observation. In logic if you begin with a false premise (in this case that the stated conditions are true) then you can logically arrive at any conclusion. So there may be many seemingly correct answers to the question as contradictorily stated depending upon which approach is taken to solving the problem. There may even be a way to use the contradiction to arrive at a bogus solution of 650, but you should not care. $\endgroup$ – John Wayland Bales May 20 '16 at 13:33

From the right angled triangle directly, connect two triangles in horizontal plane:

$$ 1515 \sqrt{ \cot ^2 31^0 - \cot^2 35^0} \approx 1294 $$

  • $\begingroup$ If you look at my question I got 1294 as one of my answers but I also got 460 using the same information. $\endgroup$ – user262291 May 20 '16 at 1:11
  • $\begingroup$ The $258^0$ elevation information at end of tunnel is redundant , inconsistent or unnecessary. $\endgroup$ – Narasimham May 20 '16 at 1:21

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