Question 1151498
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Plane #1 is traveling 800 km/hr with bearing *[Tex \Large \text{S}35^{\circ}\text{E}]
Plane #2 is traveling 750 km/hr with bearing *[Tex \Large \text{S}25^{\circ}\text{W}]


Both planes take off at 12:00 pm noon, and they both take off from point A. 
At 2:30 pm, plane #1 arrives at point B and plane #2 arrives at point C. 
The flight duration so far is 2.5 hours.


Diagram
<img src = "https://i.imgur.com/ClEhxph.png">
The notation *[Tex \Large \text{S}35^{\circ}\text{E}] means we start looking directly south and then turn 35 degrees toward the east as the diagram above shows for the blue angle. 
The red angle is set up in a similar fashion. 
For more information about compass bearings, see this page below.
<a href = "http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html">http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html</a>


The red and blue angles combine to get 25+35 = 60 which is the green angle. This is angle A.


The sides b and c are adjacent to angle A. 
side b is opposite angle B
side c is opposite angle C
The lowercase letters are used for side lengths; the uppercase letters for angles.


From the diagram:
A = 60
b = 1875
c = 2000
The values 1875 and 2000 are the distances each plane travels for their given speeds, and when the elapsed time is t = 2.5
Plane #1 travels at 800 km/hr for 2.5 hrs, so d = r*t = 800*2.5 = 2000
Plane #2 travels at 750 km/hr for 2.5 hrs, so d = r*t = 750*2.5 = 1875


The goal is to find the length of side 'a'. Use the law of cosines here
a^2 = b^2 + c^2 - 2*b*c*cos(A)
a^2 = 1875^2 + 2000^2 - 2*1875*2000*cos(60)
a^2 = 3515625 + 4000000 - 7500000*cos(60)
a^2 = 3515625 + 4000000 - 7500000*0.5
a^2 = 3515625 + 4000000 - 3750000
a^2 = 3765625
a = sqrt(3765625)
a = 1940.5218370325
a = 1940.52


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Answer
At 2:30 pm, the two planes are roughly <font color=red size=4>1940.52 km</font> apart.
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