Question 327992
Jan is flying a plane on a triangular course at 320 mi/h. She flies due east for two hours and then turns right through a 65 degree angle. How long after turning will she be exactly southeast of where she started?
<pre><b>
Let the black line in the drawing below represent her path.  
She starts out at A, flies for 2 hours at 320 mi/h from A to B.  
That's 640 miles, so AB = 640 miles. Then she turns at a 65° 
angle and flies to C.  The problem asks to find the time it took 
her to fly from B to C.

{{{drawing(400,400,-100,1250,-1250,100, locate(0,60,A),locate(640,60,B),
locate(1200,-1200,C), locate(260,0,640miles),
green(line(640,0,1000,0)), locate(690,-20,"65°"),
line(0,0,640,0), line(640,0,1199.192774,-1199.192774), 
green(arc(640,0,300,-300,295,360))
)}}}

Now in order for point C to be exactly southeast of A, the red line
AC must make a 45° with AB, so we have this:

{{{drawing(400,400,-100,1250,-1250,100, locate(0,60,A),locate(640,60,B),
locate(1200,-1200,C), locate(260,0,640miles),
green(line(640,0,1000,0)), locate(690,-20,"65°"),
line(0,0,640,0), line(640,0,1199.192774,-1199.192774), 
green(arc(640,0,300,-300,295,360)),
red(line(0,0,1199.192774,-1199.192774), arc(0,0,300,-300,315,360)),
locate(55,-12,"45°")

)}}}

Angle ABC is supplementary to the 65°-angle, so it's 180°-65°=115°.
Since the three angles of triangle ABC must have sum 180° we can
calculate angle C as 180°-(45°+115°) = 20°, so we indicate the
measures of those two angles:

{{{drawing(400,400,-100,1250,-1250,100, locate(0,60,A),locate(640,60,B),
locate(1200,-1200,C), locate(260,0,640miles),
green(line(640,0,1000,0)), locate(690,-20,"65°"),
line(0,0,640,0), line(640,0,1199.192774,-1199.192774), 
green(arc(640,0,300,-300,295,360)), locate(550,-24,"115°"),
red(line(0,0,1199.192774,-1199.192774), arc(0,0,300,-300,315,360)),
locate(55,-12,"45°"), red(arc(640,0,250,-250,180,295)),
red(arc(1199.192774,-1199.192774,800,-800,115,135)),
locate(960,-900,"20°")
)}}}

What we're looking for is how long it took her to fly from 
B to C.  So let's first calculate BC using the law of sines:

{{{BC/sin(A)}}}{{{""=""}}}{{{AB/sin(C)}}}

{{{BC/sin("45°")}}}{{{""=""}}}{{{640/sin("20°")}}}

{{{BC}}}{{{""=""}}}{{{(640sin("45°"))/sin("20°")}}}{{{""=""}}}{{{1323.162828miles}}}.

Now we use {{{TIME}}}{{{""=""}}}{{{DISTANCE/RATE}}} to find 
the time it took her to fly the 1323.162828 miles from B to C.

{{{TIME}}}{{{""=""}}}{{{DISTANCE/RATE}}}{{{""=""}}}{{{1323.162828/320}}}{{{""=""}}}{{{4.134883836hours}}} or about 4 hours and 8 minutes. 

Edwin</pre>