City A is 300km east of city B, city C is 200km on a bearing of 123° from city B. how far is city C from city A
442
Given: BC = 200km[123o], BA = 300km[90o]. AC = ?
AC = 300km[90o]+200km[123o]
AC = (300*sin90+200*sin123)+(300*cos90+200*cos123)i
AC = 252-109i = 275kn[-67o] = 275km[-67o] = 275km[293o]CW.
To find the distance between City A and City C, we can use trigonometry and vector addition.
First, let's draw a diagram to visualize the problem. Assume City B is at the origin (0,0) on a coordinate plane, and let's place City A to the east and City C at a bearing of 123° from City B.
Now, let's break down the problem step by step:
1. Find the x-coordinate and y-coordinate of City C:
- Since City C is at a bearing of 123° from City B, we can use the cosine and sine functions to calculate its x-coordinate and y-coordinate respectively.
- x-coordinate of City C (C_x) = City B's x-coordinate (B_x) + distance from City B to City C * cos(123°)
- y-coordinate of City C (C_y) = City B's y-coordinate (B_y) + distance from City B to City C * sin(123°)
2. Calculate the x-coordinate and y-coordinate of City A:
- Since City A is 300 km east of City B, its x-coordinate will be City B's x-coordinate + 300.
- The y-coordinate will remain the same since City A is not north or south of City B.
3. Calculate the distance between City A and City C using the distance formula:
- distance between City A and City C = square root of [(C_x - A_x)^2 + (C_y - A_y)^2]
Let's perform the calculations:
1. Calculate City C's coordinates:
- B_x = 0 (City B is the origin on the x-axis)
- B_y = 0 (City B is the origin on the y-axis)
- C_x = B_x + distance from City B to City C * cos(123°)
- C_x = 0 + 200 km * cos(123°)
- C_x ≈ -117.496 km
- C_y = B_y + distance from City B to City C * sin(123°)
- C_y = 0 + 200 km * sin(123°)
- C_y ≈ 150.288 km
2. Calculate City A's coordinates:
- A_x = B_x + 300 km
- A_x = 0 + 300 km
- A_x = 300 km
- A_y = B_y + 0 km (since City A is east of City B)
- A_y = 0 + 0 km
- A_y = 0 km
3. Calculate the distance between City A and City C:
- distance between City A and City C = sqrt((C_x - A_x)^2 + (C_y - A_y)^2)
- distance between City A and City C = sqrt((-117.496 km - 300 km)^2 + (150.288 km - 0 km)^2)
- distance between City A and City C ≈ 383.667 km
Therefore, City C is approximately 383.667 km away from City A.