Question 1195044: A plane flies 75 miles south from an airfield, and then travels 120 miles more in a different direction. By this time, it bears S29°E of the airfield. In what direction is it heading?
Found 2 solutions by math_tutor2020, Alan3354:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
This is what the diagram could look like
I used GeoGebra to make the diagram. It is a free app.
Points:
A = airfield's location
B = location 75 miles south of point A
C = plane's current location
Segments:
AB = 75
BC = 120
Angle:
The notation S29°E means "start facing directly south. Then turn 29 degrees to the east".
This is why angle BAC (aka angle A) is 29 degrees.
The segment AC is in the southeast quadrant (imagine you placed the airfield at the origin)
We need to find angle ACB, or angle C for short.
Use the law of sines
sin(A)/a = sin(C)/c
sin(29)/120 = sin(C)/75
sin(C) = 75*sin(29)/120
sin(C) = 0.303006
C = arcsin(0.303006) or C = 180-arcsin(0.303006)
C = 17.638240 or C = 162.361760
These results are approximate.
Make sure your calculator is in degree mode.
If C = 17.638240, then,
A+B+C = 180
B = 180-(A+C)
B = 180-(29+17.638240)
B = 133.36176
If C = 162.361760, then,
B = 180-(A+C)
B = 180-(29+162.361760)
B = -11.36176
A negative angle measure isn't possible, so we ignore C = 162.361760 and go with C = 17.638240 only.
When C = 17.638240 degrees approximately, we found that angle B is approximately 133.36176 degrees.
The 133.36176 degrees is when we're situated at B and facing north. Turning 133.36176 eastward will have the person at B face toward C.
So one possible answer could be to say N 133.36176° E
However, the convention is to have the bearing angle to be between 0 and 90. This way we can confine it to a specific quadrant.
Note that 180-B = 180-133.36176 = 46.63824
So let's place ourselves at point B again, but this time face directly south. Then turn roughly 46.63824 degrees to the east.
This will have you aim at point C, and it will determine the plane's heading.
The plane is headed toward the bearing of S 46.63824° E
Let's round to the nearest whole number degree to get S 47° E as our final answer.
I rounded this way because the S29°E mentioned 29 being a whole number.
If your teacher instructs some other level of rounding precision, then be sure to follow said instructions.
Side note: We use segment BC to help us determine the plane's heading or bearing. This is because the plane is traveling along this segment (assuming it does not turn). It might be tempting to use segment AC instead of BC, but that would be incorrect because the plane is not headed along the direction of AC.
Answer by Alan3354(69443)  (Show Source):
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