An airplane flies 137 km due west from city A to city B and then 412 km in the direction of 46.5◦ north of west from city B to city C. What is the distance between city A and

city C?
Relative to city A, in what direction is city C? Answer with respect to due east, with the counter-clockwise direction positive, within the limits of −180
◦to +180◦.

thanks

jigga

To find the distance between city A and city C, we can use the Pythagorean theorem because the path from A to C forms a right triangle with the path from A to B and the path from B to C.

First, we need to find the horizontal distance traveled from A to C. This can be calculated by finding the sum of the horizontal distances from A to B and from B to C.

The horizontal distance from A to B is 137 km due west.

Next, we need to find the vertical distance traveled from B to C. This can be calculated using trigonometry. The angle formed between the direction from B to C and due west is given as 46.5 degrees north of west. To find the vertical component, we can use the equation:

Vertical distance = Horizontal distance * sin(angle)

The horizontal distance from B to C is 412 km, and the angle is 46.5 degrees north of west. So, the vertical distance from B to C is:

Vertical distance = 412 km * sin(46.5 degrees)

Now that we have the horizontal and vertical distances, we can find the total distance using the Pythagorean theorem:

Distance = √(horizontal distance^2 + vertical distance^2)

Substituting the values we found:

Distance = √((137 km)^2 + (412 km * sin(46.5 degrees))^2)

Evaluating this expression will give us the distance between city A and city C.

To find the direction of city C with respect to city A, we can use trigonometry. The angle formed between the path from A to B and the direction from A to C can be calculated using the equation:

Angle = sin^(-1)(vertical distance / distance)

Substituting the values we found:

Angle = sin^(-1)((412 km * sin(46.5 degrees)) / distance)

Solving this equation will give us the angle in degrees. This angle can be interpreted as the direction of city C from city A, where due east is 0 degrees, and angles increase counterclockwise.

By following these steps and performing the necessary calculations, you will be able to find the distance between city A and city C, as well as the direction of city C relative to city A.