3D Trig Question?

Question

I've been having trouble with this question:

David is in a life raft and Anna is in a cabin cruiser searching for him. They are in contact by mobile telephone. David tells Anna that he can see Mt Hope. From David's position the mountain has a bearing of $109$ degrees, and the angle of elevation to the top of the mountain is $16$ degrees. Anna can also see Mt Hope. From her position it has a bearing of $139$ degrees, and the top of the mountain has an angle of elevation of $23$ degrees. The top of the Mt Hope is $1500$m above sea level.

Find the distance and bearing of the life raft from Anna's position.

Any help would be appreciated, Thanks

Answer 1

Going to assume you are treating the ocean as a flat surface! Get the distane of each person from the base (sea-level) of the mountain using simple trig with the tangent function. (Approx 3534m and 5231m).

Now starting from the base of the mountain, draw bearings to each person by reversing the bearings given (add 180). Add the distances to your diagram. You should get a triangle with sides 3534, 5231 and an angle of 30 degrees in between (by considering your two reversed bearings). Finally use the Cosine rule to find the unknown side which is the distance between them. Then use the Sine rule to find all angles in the triangle, from which you will be able to work out the relevant bearing.

Answer 2

This is not really a 3D geometry problem. You can see this by realizing that David, Anna, and the base of Mt. Hope form a triangle. The height of Mt. Hope, which I'll call $H$, and the the angles of elevation $\theta_D$ and $\theta_A$, determine the lengths of two of the sides of the triangle as $H \cot{\theta_D}$ and $H \cot{\theta_A}$. Now we have a 2D geometry problem, and all we need to determine the desired distance between David and Anna is to determine the length of the third side of the triangle, which means that we need the angle between the two known sides.

Here is a picture:

Hopefully it is clear that $|MD| = H \cot{\theta_D}$ and $|MA| = H \cot{\theta_A}$. It should also be clear that the angle we need is $\angle DMA =\phi_A-\phi_D$, or the difference between the bearings. To see this, form the triangle $\Delta D'MA$ and note that $\phi_D + \angle D'MA + \pi-\phi_A=\pi$.

The distance $|DA|$ between David and Anna is then found from the law of cosines:

$$|DA|^2 = H^2 \cot^2{\theta_A} + H^2 \cot^2{\theta_D} - 2 H^2 \cot{\theta_A} \cot{\theta_D} \cos{(\phi_A-\phi_D)}$$

From the given numbers, I get that the distance between David and Anna is about $2799$ m.

Answer 3

Note_1. If the bearing is given in 3-digit form, true bearing is assumed. Thus the figure should be re-drawn as figure 1 of

Note_2. If sine law [sin DAB = (5231 / 2799) sin 30] is used to find the angle DAB, caution must be taken because the calculator will only return (69 degrees) -- the smaller of the possible solution. However, the actual angle DAB should be (180 – 69 = 111 degrees) [See figure 2.]. It is saver to use cosine law to find angle DAB instead.