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?
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?
Best Answer
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.