An airplane flies 200 km due west from city A to City B and them 300 km in the direction of 30.0 degrees north of west from city B to city C. In straight-line distance, how far is city C from city A? Relative to city A, in what direction is city C?

Calculate the cumulative coordinates of B and C relative to A by resolving the distances into X(east) and Y(north) components.

From A to B, distance Db = 200 km,
direction = 180 degrees.
x=Db.cos(180)=-200
y=Db.sin(180)=0
Therefore the coordinates of B relative to A are (-200,0).

From B to C:
Distance Dc = 300 km
direction = 180-30=150 degrees.
x=300 cos(150)=-259.81 km
y=300 sin(150)=150 km
The coordinates of C relative to B are (-259.81, 150).

The cumulative coordinate relative to A are
(-200+(-259.81), 0+150)
=(-459.81, 150)

Can you now calculate the distance and direction?

To find the straight-line distance between City C and City A, you can use the Pythagorean theorem because the airplane's path forms a right triangle.

First, let's break down the distances travelled by the airplane:

From City A to City B: 200 km due west
From City B to City C: 300 km at an angle of 30.0 degrees north of west

To find the distance from City A to City C, we need the lengths of the two sides of the right triangle formed:

Side AB (opposite to the right angle): 200 km
Side BC (adjacent to the 30.0 degrees angle): 300 km

Using the Pythagorean theorem, which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, we can calculate the distance from City A to City C:

AC² = AB² + BC²

AC² = 200² + 300²
AC² = 40000 + 90000
AC² = 130000

AC = √130000
AC ≈ 360.55 km

Therefore, the straight-line distance from City C to City A is approximately 360.55 km.

To determine the direction of City C relative to City A, we can use trigonometry. We know that the airplane flew 200 km due west from City A, and then 300 km at an angle of 30.0 degrees north of west from City B.

The angle between the line connecting City A and City C and the due west line (from City A) can be found by subtracting 30.0 degrees from 180 degrees (because the angle is measured from the west direction). Therefore:

Angle from City A to City C = 180 degrees - 30.0 degrees
Angle from City A to City C ≈ 150.0 degrees

So, relative to City A, City C is approximately 360.55 km away, in the direction of 150.0 degrees.