A car traveling at a constant rate due east passes an interstate exit that is 1.0 mile from a gas station. The

gas station is S50°E of the exit. After the car has traveled for 2.0 minutes, its bearing to the gas station is
S55°W. How fast is the car traveling?

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.

To find the speed of the car, we can use the concept of velocity. Velocity is defined as the rate of change of displacement with respect to time. In this case, we need to find the rate at which the car is traveling, which is essentially the speed.

Let's break down the given information:
1. The car is traveling at a constant rate due east.
2. The distance between the interstate exit and the gas station is 1.0 mile.
3. The gas station is located at a bearing of S50°E from the exit.
4. After traveling for 2.0 minutes, the car's bearing to the gas station is S55°W.

Here's how we can approach this problem:

Step 1: Convert the bearing angles into compass bearings:
- S50°E can be converted to (180° - 50°) = 130°.
- S55°W can be converted to (180° + 55°) = 235°.

Step 2: Determine the difference in bearing angles:
To find the change in bearing from the first to the second position, we subtract the initial bearing from the final bearing:
235° - 130° = 105°.

Step 3: Calculate the displacement:
The displacement is the shortest straight-line distance between two points. In this case, it is the distance from the exit to the gas station. Since the car is traveling due east, the displacement is a horizontal line, and we can use basic trigonometry to calculate it. We can use the cosine function because it relates the adjacent side length to the hypotenuse in a right triangle:
cos(130°) = adjacent/hypotenuse.
The adjacent side represents the displacement we need to find, and the hypotenuse represents the 1.0-mile distance between the exit and the gas station.
Let's solve for the adjacent side:
adjacent = cos(130°) * 1.0 mile.

Step 4: Determine the time and calculate speed:
We know that the car has traveled for 2.0 minutes. Speed is defined as the distance traveled divided by the time taken. So, we can find the speed using the equation:
speed = displacement/time.
Since the displacement is given in miles and the time is given in minutes, we should ensure the units are consistent.

Now let's put it all together:

- Convert the bearing angles:
- S50°E = 130°
- S55°W = 235°

- Determine the difference in bearing angles:
235° - 130° = 105°

- Calculate the displacement:
adjacent = cos(130°) * 1.0 mile

- Determine the time and calculate speed:
speed = displacement/time

Now you can substitute the relevant values into the equations to find the answer.