An airplane flies 200 km due west from city A to city B and then 245 km in the direction of 31.5° north of west from city B to city C.

(a) In straight-line distance, how far is city C from city A?

To find the straight-line distance between city A and city C, we can use the Pythagorean theorem.

The first step is to find the east-west component of the distance between city B and city C. Since the airplane flew 245 km in the direction 31.5° north of west, we can use trigonometry to find the east-west component. To do this, we need to find the adjacent side of the triangle formed by the distance and angle.

Adjacent side = distance * cos(angle) = 245 km * cos(31.5°)

Next, we need to find the north-south component of the distance between city B and city C. Since the airplane flew 245 km in the direction 31.5° north of west, we can use trigonometry to find the north-south component. To do this, we need to find the opposite side of the triangle formed by the distance and angle.

Opposite side = distance * sin(angle) = 245 km * sin(31.5°)

Now, we can find the overall east-west distance by subtracting the east-west component traveled from city B to C from the 200 km traveled from city A to B.

East-West distance = 200 km - (245 km * cos(31.5°))

Finally, to find the straight-line distance between city A and city C, we can use the Pythagorean theorem.

Distance = sqrt((East-West distance)^2 + (North-South distance)^2)

Plug in the values and calculate to find the answer.