City A is 300kilometer due east of city B. city C is 200kilometer on a bearing of 123degree from city B.how far is it from C to A
Note that angle ABC = 33 degrees. So, using the law of cosines, your distance d is given by
d^2 = 300^2 + 200^2 - 2(300)(200) cos33
plug and chug
Gat
64.857
Please I want the diagram
To find the distance from city C to city A, we can use the concept of vectors. First, let's break down the given information:
City A is 300 kilometers due east of city B.
City C is 200 kilometers on a bearing of 123 degrees from city B.
Now, let's draw a diagram to visualize the situation:
B
/|
/ |
/ |
/ |
300km | 200km
A-------C
In the diagram, city B is located at the origin of a coordinate system. City A is 300 kilometers due east, so we can represent its position as (300, 0). City C is 200 kilometers on a bearing of 123 degrees from city B, which means we need to find the coordinates relative to B and then add them to B's coordinates.
To find the coordinates of C relative to B, we can use trigonometry. The bearing angle of 123 degrees can be translated into an angle from the positive x-axis (east direction). Since the bearing starts from the positive y-axis (north direction), we subtract 90 degrees from it.
123 degrees - 90 degrees = 33 degrees
Using the angle of 33 degrees and the distance of 200 kilometers, we can find the coordinates relative to B using trigonometry:
x-coordinate: 200 * cos(33 degrees) ≈ 166.40 kilometers
y-coordinate: 200 * sin(33 degrees) ≈ 107.08 kilometers
Now, we can add these coordinates to B's coordinates:
C's coordinates = (B's x-coordinate + C's relative x-coordinate, B's y-coordinate + C's relative y-coordinate)
C's coordinates = (0 + 166.40, 0 + 107.08)
C's coordinates ≈ (166.40, 107.08)
Finally, to find the distance from C to A, we can use the distance formula between two points in a 2D space:
Distance = √[(xA - xC)^2 + (yA - yC)^2]
Plugging the coordinates of A and C into the formula:
Distance = √[(300 - 166.40)^2 + (0 - 107.08)^2]
Distance ≈ √[133.60^2 + (-107.08)^2]
Distance ≈ √[17856 + 11475.0464]
Distance ≈ √29331.0464
Distance ≈ 171.18 kilometers
Therefore, the distance from city C to city A is approximately 171.18 kilometers.