The bearing from City A to City B is N $41^{\circ} \mathrm{E}$. The bearing from City B to City C is S $49^{\circ} \mathrm{E}$. An automobile driven at $55 \mathrm{mph}$ takes 1.2 hours to drive from City A to City B and takes 1.4 hours to drive from City B to City C. Find the distance from City A to City C. (Neglect the curvature of the earth.)
The distance from City A to City $\mathrm{C}$ is approximately miles. (Round to the nearest mile as needed.)

Step 1 ：Given that the speed of the automobile is 55 mph, and it takes 1.2 hours to travel from City A to City B, we can calculate the distance from City A to City B as \(55 \times 1.2 = 66\) miles.

Step 2 ：Similarly, given that it takes 1.4 hours to travel from City B to City C, we can calculate the distance from City B to City C as \(55 \times 1.4 = 77\) miles.

Step 3 ：The bearing from City A to City B is N $41^{\circ} \mathrm{E}$, and the bearing from City B to City C is S $49^{\circ} \mathrm{E}$. These bearings form an angle of \(90^{\circ}\) at City B.

Step 4 ：Using the law of cosines, we can calculate the distance from City A to City C as \(\sqrt{66^2 + 77^2 - 2 \times 66 \times 77 \times \cos(90^{\circ})}\), which simplifies to approximately 101 miles.

Step 5 ：Final Answer: The distance from City A to City C is approximately \(\boxed{101}\) miles.