Calculate the height of a distant object using estimated angles from two different points.

$\begingroup$

I knew how to do this long ago, found the exact problem in my old trig book, but I can't seem to work it out.

Say I'm at an unknown distance from a mountain, called point P, and I estimate the angle of elevation to the top of the mountain is 13.5 degrees. Then I move to point N, which is 100 meters closer to the mountain, and I estimate the angle of elevation to be 14.8 degrees. What is the height of the mountain?

I remember this being enough information to solve both triangles, but without the distance to the mountain, or the height of the mountain, I'm at a loss. Hint's would be appreciated.

$\endgroup$ 1

5 Answers

$\begingroup$

Or...

Use the Law of Sines to find the longest side in the triangle with the 100 m side (you know all the angles).

This longest side is also the hypotenuse of a another, right angle triangle where you know the angle opposite the height you want...

$\endgroup$ 1 $\begingroup$

This picture will give you a good idea. Use the $\tan$ function on both angles, and solve the equation because

$$ \begin{align*} \frac{height}{length+100} &= \tan 13.5^\circ \\ \frac{height}{length} &= \tan 14.8^\circ \end{align*} $$

It seems that you will get a pair of simultaneous equations. So 2 linear equations and 2 unknowns, pretty easy to solve.

I will respond to your comment right here. What you can do is to solve by cancelling height.

Eg.$$ \begin{align*} \\tan 14.8^\circ \times {length} &= \tan 13.5^\circ \times ({length+100}) \end{align*} $$There are other ways to solve the linear simultaneous equations. I'll leave it to you to figure them all out.

$\endgroup$ 2 $\begingroup$

Let $h$ be the height of the mountain (in meters), and $d$ the distance from $P$ to the mountain. Then you have $h/d=\tan(13.5)$, and $h/(d-100)=\tan(14.8)$ (all angles in degrees), which you can solve for $h$ and $d$.

$\endgroup$ 2 $\begingroup$

Here's what I've come up with:

α = angle of elevation at P = 13.5 deg
β = angle of elevation at N = 14.8 deg
d = distance between points P and N = 100m
h = height of mountain
h = (d * tan β * tan α) / (tan β - tan α)
h = (100 * tan 14.8 * tan 13.5) / (tan 14.8 - tan 13.5) = 262.8m

I double checked this answer by finding the intersection of bryansis2010's two equations:

x * tan 14.8 = (x + 100) * tan 13.5

$\endgroup$ $\begingroup$

Let $O$ and $M$ the base point and the tip of the mountain, respectively. Then $$NM=100\cdot\frac{\sin13.5}{\sin1.3},$$ and $OM$ is the height of the mountain, where $$OM=NM\cdot\sin14.8\approx262.854.$$ Note that the input of the sine is in degrees.

$\endgroup$