City A is 300km due east of city B ..city C is 200km on a bearing of 123* from city B how far is it from C to A
use the law of cosines.
To find the distance from City C to City A, we can use the concept of vectors and trigonometry. Let's break down the problem step by step:
Step 1: Calculate the east-west distance between City B and City C.
- Since City A is due east of City B, there is no east-west component between them.
- City C is at a bearing of 123° from City B. We can visualize this as an angle measuring counterclockwise from the eastward direction.
- The distance between City B and City C is given as 200km.
Step 2: Convert the bearing to a direction vector.
- A direction vector can be represented as (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis.
- In our case, the bearing of 123° can be converted to a direction vector as follows:
- x-component: cos 123°
- y-component: sin 123°
Step 3: Calculate the north-south and east-west components of the direction vector.
- Since the direction vector only represents the bearing from City B, we need to calculate the actual distances in the north-south and east-west directions.
- north-south component: y-component of the direction vector × distance between City B and City C
- east-west component: x-component of the direction vector × distance between City B and City C
Step 4: Calculate the total east-west distance between City A and City C.
- Since City A is due east of City B, the east-west distance remains the same from City B to City C.
- Therefore, the east-west distance between City A and City C is the same as the east-west distance between City B and City C.
Step 5: Calculate the total distance between City A and City C.
- The total distance can be calculated using the Pythagorean theorem:
- (east-west distance)^2 + (north-south distance)^2 = (total distance)^2
Let's perform the calculations step by step:
Step 1:
- East-west distance between City B and City C: 0km
- Distance between City B and City C: 200km
Step 2:
- x-component of the direction vector: cos 123°
- y-component of the direction vector: sin 123°
*Note: The trigonometric functions should be calculated in radians, not degrees.
Step 3:
- North-south component: sin 123° × 200km
- East-west component: cos 123° × 200km
Step 4:
- East-west distance between City A and City C: East-west component calculated in step 3
Step 5:
- Total distance between City A and City C: √((east-west distance)^2 + (north-south distance)^2)
By following these steps and performing the necessary calculations, you will find the distance from City C to City A.