An airplane flies 200 km due west from city A to city B and then 305 km in the direction of 30.0° north of west from city B to city C.
(a) In straight-line distance, how far is city C from city A?
___km
(b) Relative to city A, in what direction is city C?
___° north of west
To solve this problem, we can use the Pythagorean theorem and trigonometric functions to find the straight-line distance and direction of city C from city A.
(a) To find the straight-line distance between city A and city C, we need to find the total displacement in both the horizontal and vertical directions.
The horizontal displacement from A to B is 200 km.
The vertical displacement from B to C is 305 km * sin(30°) = 305 km * 0.5 = 152.5 km.
Using the Pythagorean theorem, we can find the straight-line distance between A and C:
Distance = √(horizontal displacement)^2 + (vertical displacement)^2
Distance = √(200 km)^2 + (152.5 km)^2
Distance = √(40000 km^2 + 23256.25 km^2)
Distance = √(63256.25 km^2)
Distance ≈ 251.5 km
Therefore, city C is approximately 251.5 km away from city A.
(b) To find the direction of city C with respect to city A, we need to determine the angle θ between the displacement vector from A to C and the west direction.
Using trigonometry, we can find the angle θ:
θ = atan(vertical displacement / horizontal displacement)
θ = atan(152.5 km / 200 km)
θ ≈ 36.9°
Since the angle is measured clockwise from the west direction, we can say that city C is approximately 36.9° north of west with respect to city A.
To find the straight-line distance between city A and city C, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the distance from city A to city B is the first side, the distance from city B to city C is the second side, and the straight-line distance from city A to city C is the hypotenuse.
(a) First, let's calculate the distance from city A to city C:
Distance from city A to city B = 200 km
Distance from city B to city C = 305 km
Now, we can calculate the straight-line distance using the Pythagorean theorem:
Straight-line distance^2 = (Distance from city A to city B)^2 + (Distance from city B to city C)^2
Straight-line distance^2 = 200^2 + 305^2
Straight-line distance^2 = 40000 + 93025
Straight-line distance^2 = 133025
Taking the square root of both sides:
Straight-line distance = √133025
Straight-line distance ≈ 365 km
Therefore, city C is approximately 365 km from city A.
(b) To determine the direction of city C relative to city A, we can use trigonometry. The angle of 30.0° north of west indicates that city C is 30.0° above the west direction.
Therefore, relative to city A, city C is 30.0° north of west.