To solve this problem, we need to use trigonometry and geometry principles.
Step 1: Convert the given angles to the corresponding bearings in terms of degrees. The bearing S50°E can be converted to 130° clockwise from north. Similarly, the bearing S55°W can be converted to 235° clockwise from north.
Step 2: Draw a diagram based on the given information. Label the exit, gas station, and the traveled distance as shown:
Exit (E) Gas Station (G)
. . . . . . . . . . . . C (Car's position after 2.0 minutes)
Step 3: Let's assume the car's distance from the gas station after 2.0 minutes is x miles. From the given information, we know that the car has traveled x + 1.0 miles in 2.0 minutes.
Step 4: Use the Law of Cosines to find the distance between the exit and the gas station:
(x + 1.0)^2 = 1.0^2 + ?^2 – 2(1.0)(?)cos(130°)
Step 5: Simplify the equation:
x^2 + 2x + 1 = 1 + ?^2 – 2?cos(130°)
Step 6: Use the Law of Cosines again to find the distance between the car and the gas station:
?^2 = x^2 + ?^2 – 2x?cos(235°)
Step 7: Substitute the value of ?^2 from the previous equation:
x^2 + 2x + 1 = x^2 + x^2 + 2x?cos(235°) – 2x?cos(130°)
Step 8: Rearrange the equation and simplify:
2x?cos(130°) – 2x?cos(235°) = 1
Step 9: Combine like terms and solve for x:
2x[cos(130°) – cos(235°)] = 1
2x[cos(130°) + cos(55°)] = 1
Step 10: Substitute the values of cos(130°) and cos(55°):
2x[(-0.342) + 0.574] = 1
2x(0.232) = 1
Step 11: Solve for x:
0.464x = 1
x = 1/0.464
x ≈ 2.155
Step 12: Since the car's distance from the gas station after 2.0 minutes is x miles, we can conclude that the car's speed is approximately 2.155 miles per minute.
Therefore, the car is traveling at a speed of approximately 2.155 miles per minute.