Could somebody help me with this math problem about bearings?

I tried to draw the diagram in MS Word, which didn't work, so now I'll describe what I would have done on paper. I use the short expression cos rule for the rather grander one you use. Cos rule and Law of cosines mean the same thing.

Make a point in the middle of the page to represent the airport. Draw a short line straight up the paper to represent the direction North and write an N at the end of it.

When you measure bearings, you start at North, which is 0 degrees and turn clockwise until you have turned the number of degrees of the bearing.

As "there are 360 degrees in a circle", a bearing of 355 degrees is 5 degrees short of returning to N (in other words the bearing is 5 degrees West of "or to the left of" N).

Put the centre of the protractor on the airport as this is where the angles are being measured. (If you haven't got a protractor, just guess the angles.) You're going to calculate the answer any way. There's no doubt that an approximately realistic diagram helps.

You want to draw a line 5 degrees to the west of the short line representing North. The length of this line should represent the speed of the first aeroplane. Draw the line and write 425 mph next to it.

The second aeroplane leaves the airport at a bearing of 67 degrees. So draw a line that looks about 67 degrees clockwise from the short North line running through the airport. Remember that this line represents a plane travelling at 530 mph, so make that line a bit longer than the one representing the speed of the first aeroplane.

You now have a point representing the airport and 2 lines representing the speeds of the planes in miles per hour.

If a plane is travelling at 300 mph, it travels 300 miles in 1 hour.

So you could tell me how many miles each of those planes in the question travels in one hour.

Those lines you've drawn also represent the distance the planes have flown in 1 hour.

Now draw a third line joining the positions of the 2 planes after 1 hour of flight. This 3rd line represents the distance between the 2 aeroplanes after 1 hour of flight. This diagram represents a snapshot of the positions of the planes after 1 hour of flight

You now have a triangle. You know the lengths of 2 of the sides. (You know how many miles the planes have flown in 1 hour.)

You also know he size of the angle between them. When you've done the drawing, you'll see it is 5 degrees + 67 degrees = 72 degrees.

If you know 2 sides and the angle between them, you can use the cos rule to solve it.

a squared = b squared + c squared - 2bc cosA,

where a is the distance between the 2 planes after 1 hour and A is the angle between the two tracks of the planes startin at the airport.

Nearly finished now! Only one more step.

You have found the distance between the planes after 1 hour of flight. The question wants to know the distance between them after 2 hours of flight.

Since the speeds and directions are constant, a 2 hour flight will produce a similarly shaped triangle that has each side twice as big.

The planes will have flown 2 x 425 miles and 2 x 530 miles and the distance between them after 2 hours will be twice the value you got for the solution to "a" in the cos rule (measured of course in miles per hour).

Phew - time for a cup of tea.

Seriously, it's very easy as soon as you have a diagram and if you remember the cos rule.

I'm curious too

Funny, I was wondering the same thing myself