An airplane leaves airport A and flies 200 km. At this time its bearing from Airport B, 250 km to the west, is 120 degrees. How far is the airplane from B?
To determine the distance of the airplane from Airport B, we can visualize the situation and use trigonometry.
Step 1: Visualize the situation
Draw a diagram to represent the situation. Place airports A and B on the diagram, with a distance of 250 km between them. Mark the airplane's location after flying 200 km from Airport A.
Step 2: Analyze the geometry
The situation can be represented as a triangle. Let's name the airplane's final location as point C. We know the distance between airports A and B (250 km), the distance from airport A to point C (200 km), and the angle at airport B (120 degrees).
Step 3: Use the Law of Cosines
The Law of Cosines states that in a triangle with sides of lengths a, b, and c, and an angle opposite side c, we have the following relationship:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, side a (250 km) and side b (200 km) are known, as well as angle C (120 degrees). We want to find side c (the distance from point C to airport B).
Substituting the known values into the formula, we have:
c^2 = (200 km)^2 + (250 km)^2 - 2 * 200 km * 250 km * cos(120 degrees)
Step 4: Solve for c
Calculate c using the given formula:
c^2 = 40000 km^2 + 62500 km^2 - 2 * 200 km * 250 km * cos(120 degrees)
c^2 = 102500000 km^2 - 100000 km * cos(120 degrees)
Now we can solve for c by finding the square root of both sides:
c = sqrt(102500000 km^2 - 100000 km * cos(120 degrees))
Plugging in the values and calculating:
c ≈ 277.13 km
Therefore, the airplane is approximately 277.13 km away from Airport B.