PLEASE HELP ME!!
The diagram shows the position of three airports . A,E and G. G is 200 kilometers from A, E is 160 kilometers from A, From G the bearing of A is 052 degrees, From A the bearing of E is 216 degrees. How far apart are airports G and E?
A to G is 52 degrees below the negative x direction
A to E is 216 - 180 = 36 degrees below negative x direction
so angle AEG = 52 - 36 = 16 degrees
now law of cosines
EG^2 = 200^2 + 160^2 - 2*200*160 cos 16
See previous post: 4 - 26 -20, 7:02 AM.
To find the distance between airports G and E, you can use the angle and side relationship in a triangle. Here are the steps to solve this problem step-by-step:
1. Draw a diagram with three points: A, E, and G.
2. Label the distances:
- Distance AG = 200 km
- Distance AE = 160 km
3. Label the angles:
- Angle = 052 degrees (the bearing of A from G)
- Angle AEG = 216 degrees (the bearing of E from A)
4. Note that angles and AEG form a triangle, and we need to find the missing side.
5. Use the angle sum property of a triangle: The sum of angles in a triangle is 180 degrees.
- Angle + Angle AEG + Angle EAG = 180 degrees
- 052 degrees + 216 degrees + Angle EAG = 180 degrees
6. Solve for Angle EAG:
- 268 degrees + Angle EAG = 180 degrees
- Angle EAG = 180 degrees - 268 degrees
- Angle EAG = -88 degrees (Note: Since the angles and AEG are given, the remaining angle has to be -88 degrees to add up to 180 degrees.)
7. Use the Law of Sines to find the length of side GE:
- sin() / GE = sin(EAG) / AE
- sin(052 degrees) / GE = sin(-88 degrees) / 160 km (Note: Sine of -88 degrees is equal to the sine of its positive supplementary angle, 92 degrees)
8. Solve for GE:
- GE = (sin(052 degrees) * 160 km) / sin(92 degrees)
Calculating this value gives us:
GE ≈ 115.04 km
Therefore, airports G and E are approximately 115.04 kilometers apart.
To find the distance between airports G and E, we can use the concept of the Law of Cosines from trigonometry.
First, let's label the distances:
GA = 200 km
EA = 160 km
Now, we can use the Law of Cosines formula, which states: c² = a² + b² - 2ab*cos(C), where c is the unknown side we are trying to find.
In this case, side c is the distance between airports G and E. Side a is the distance GA, and side b is the distance EA.
Since we know the bearings, we can deduce the angle between GA and EA. To do that, we need to subtract the bearing of A from the bearing of E:
Angle = (216° - 052°).
However, the Law of Cosines requires the angle to be in radians, not degrees. So, we need to convert the angle to radians by multiplying it by (π/180):
Angle = ((216 - 52) * π/180) radians.
Now, substituting the values into the Law of Cosines formula, we get:
c² = (200)² + (160)² - 2(200)(160) * cos(Angle ).
Next, we can simplify the equation:
c² = 40000 + 25600 - 64000 * cos(Angle ).
Lastly, we can compute the value of c by taking the square root of both sides of the equation:
c = √(40000 + 25600 - 64000 * cos(Angle )).
Let's calculate the value of c:
c = √(40000 + 25600 - 64000 * cos((216 - 52) * π/180)).
Using a calculator, we can determine the final result of c, which represents the distance between airports G and E.