# Thread: Help solve confusing bearings problem

1. ## Help solve confusing bearings problem

Hi everyone,

I’m a math tutor and quite embarrassingly, I’m hahing a tough time solving this problem. There may well be a really simple solution to it but I’m just not seeing it for some reason. Any help would be greatly appreciated.

An aircraft flying at a height of 500m above ground is sighted at a point A’ due east of an observer at a point O on the ground, measured horizontally to be 1km from the plane. The aircraft is flying south-west.
a) find its bearing from O at this time? (answer is 215.65)
b) what will be it’s angle of elevation from O at this time? (Answer is 6.55)

This has me utterly baffled I’m afraid, sigh 😔

2. ## Re: Help solve confusing bearings problem

If the observer is due east of the plane the bearing to the plane is simply due west or a bearing of $\theta = 270^\circ$

There must be some missing info to obtain the answer they have.

The elevation $\phi$ is simply

$\phi = \arctan\left(\dfrac{500}{1000}\right) = 26.6^\circ$

Again there must be some missing information here.

3. ## Re: Help solve confusing bearings problem

You were absolutely right, I hadn't appreciated the question in its entirety. It also said that it was travelling 300km/h. So the questions above were to be calculated after the aircraft had travelled for 1 minute (i.e. 5km).

I'm still struggling to figure out the answer though.

4. ## Re: Help solve confusing bearings problem

Here's the full question again. Man I'm a doofus.

An aircraft flying at a height of 500m above ground is sighted at a point A’ due east of an observer at a point O on the ground, measured horizontally to be 1km from the plane. The aircraft is flying south-west at 300km/h.
a) how far has it traveled after 1 minute?
b) find its bearing from O at this time? (answer is 215.65)
c) what will be it’s angle of elevation from O at this time? (Answer is 6.55)

5. ## Re: Help solve confusing bearings problem

Here's the full question again. Man I'm a doofus.

An aircraft flying at a height of 500m above ground is sighted at a point A’ due east of an observer at a point O on the ground, measured horizontally to be 1km from the plane. The aircraft is flying south-west at 300km/h.
a) how far has it traveled after 1 minute?
b) find its bearing from O at this time? (answer is 215.65)
c) what will be it’s angle of elevation from O at this time? (Answer is 6.55)

6. ## Re: Help solve confusing bearings problem

Thank you for that but I’m atruggling to decipher the steps in the workings for this. Is it possible for you elaborate on the working out?

7. ## Re: Help solve confusing bearings problem

Thank you for that but I’m atruggling to decipher the steps in the workings for this. Is it possible for you elaborate on the working out?
what don't you understand?

8. ## Re: Help solve confusing bearings problem

How are the values of -2535.53 and -3535.53, calculated?

The hypotenuse of the triangle would be 1414.43 (1000/cos(45)). So if the plane travels for 5,000m, wouldn’t it 5,000-1414.43 = 3,585.53? What am I doing wrong there?

9. ## Re: Help solve confusing bearings problem

Originally Posted by romsek
If the observer is due east of the plane the bearing to the plane is simply due west or a bearing of $\theta = 270^\circ$
The original problem said "An aircraft flying at a height of 500m above ground is sighted at a point A’ due east of an observer at a point O on the ground" not that the observer was east of the aircraft. The bearing is $\theta= 90^\circ$, not $270^\circ$.

10. ## Re: Help solve confusing bearings problem

Originally Posted by HallsofIvy
The original problem said "An aircraft flying at a height of 500m above ground is sighted at a point A’ due east of an observer at a point O on the ground" not that the observer was east of the aircraft. The bearing is $\theta= 90^\circ$, not $270^\circ$.
yeah I know, it was fixed in the full post.

11. ## Re: Help solve confusing bearings problem

Here's a little different calculation with some pictures. Hope it helps.

I'm not a navigator, but I think bearing is a calculation in a 2 dimensional plane. The description of the problem is clearly 3 dimensional. The plane initially is at coordinates (1,0,0.5) (kilometer units) and flies southwest at constant altitude 0.5 km. So the bearing is computed in the plane z = 0.5. (Aside, I don't know how to compute a bearing in case the plane is not flying at constant altitude; I don't see what plane to use or what would be direction north.)

Next, here's a diagram that shows the angle of elevation for any point P = (x,y,z) where z is not 0 and P is not directly above the origin:

So the angle of elevation of P is
$$\arctan(0.5/\sqrt{x^2+y^2})\approx0.114=6.556\text{ degrees}$$
(Here the x and y are the previously calculated values.)

12. ## Re: Help solve confusing bearings problem

John, that was fantastic, makes perfect sense. Thank you for going through all that trouble for me

The rest of the team too