# Motion in a Plane

A car and a small plane travel from the same starting point to the same destination by diﬀerent routes. The plane travels in a straight line for 65 minutes at an average speed of 250mph (1m/s = 2.24mph). The car travels 370 miles along a route that goes due south and then due west, further west than south. If the plane travelled in a straight line, at what angle south of west did it travel?

I know I have to use the pythagorean theorem (at least i think so)
but I just cant get it set up right.

berkeman
Mentor
A car and a small plane travel from the same starting point to the same destination by diﬀerent routes. The plane travels in a straight line for 65 minutes at an average speed of 250mph (1m/s = 2.24mph). The car travels 370 miles along a route that goes due south and then due west, further west than south. If the plane travelled in a straight line, at what angle south of west did it travel?

I know I have to use the pythagorean theorem (at least i think so)
but I just cant get it set up right.

Welcome to the PF.

Start by drawing the triangle that they describe. Label the lengths of the triangle, and see if you see how to do the trig...

See I know how to do the trig, but I just cannot find the lengths of the sides other than the hypothenus

berkeman
Mentor
See I know how to do the trig, but I just cannot find the lengths of the sides other than the hypothenus

Well, you know the distance that the plane flew (as you say, the hypoteneus). So you could draw a circle centered at the origin of that radius. And the car travels 370 total miles south and then west. Can you construct a sum of the south side and west side of the triangle that totals 370 miles that intersects that circle...? Think of it algebraically...

I'm not sure if the couple of hours I have spent staring at this problem have helped me and I am probably over thinking it, but im lost.

berkeman
Mentor
I'm not sure if the couple of hours I have spent staring at this problem have helped me and I am probably over thinking it, but im lost.

Probably. How far did the plane go?

And what equation can you write relating the x and y sides of the triangle to the hypoteneuse?

You should end up with a quadratic equation, and use the fact that you are given about "farther west than south" to help you pick which of the two solutions is correct...

I got the plane to travel around 270 Miles.

As far as quadratic equations go, I was definitely not even thinking about that. All that is stuck on my paper is like x^2+y^2=c^2 with x being the west direction and y being the south.

berkeman
Mentor
I got the plane to travel around 270 Miles.

As far as quadratic equations go, I was definitely not even thinking about that. All that is stuck on my paper is like x^2+y^2=c^2 with x being the west direction and y being the south.

Good. Use 270.8 miles for the hypoteneus.

Now, you correctly write the Pythagorean theorm for the sides of the triangle. There is one other equation you need to use, which is the sum of the x and y sides equals what?

Once you have the two equations, you need to solve them together to find x and 7.

I will be offline for a while. Have at it!

Thanks for the help (sorry for being late on this). I did end up getting it, and I was really overthinking it. Thanks again :)