To find the distance from City C to City A, we first need to determine the positions of cities A, B, and C based on the information provided.
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Position of Cities:
- City B is at the origin \( (0, 0) \).
- City A is 300 km due east of City B, so its coordinates are \( (300, 0) \).
- City C is 200 km on a bearing of 123° from City B.
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Calculating the Position of City C: A bearing of 123° means that we measure 123° clockwise from the north. To convert this to standard Cartesian coordinates (where 0° is along the positive x-axis), we use the angle: \[ \theta = 123° - 90° = 33° \text{ south of east (or } 90° + 33° = 123°\text{ from north)} \]
The coordinates of City C can be found using trigonometric functions: \[ C_x = 200 \times \cos(123°) \] \[ C_y = 200 \times \sin(123°) \]
Calculating these values: \[ C_x = 200 \times \cos(123°) \approx 200 \times (-0.83867) \approx -167.734 \text{ km (to the west)} \] \[ C_y = 200 \times \sin(123°) \approx 200 \times 0.51504 \approx 103.008 \text{ km (to the north)} \]
Therefore, the coordinates of City C are approximately \( (-167.734, 103.008) \).
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Finding the Distance from City C to City A: The distance between City C and City A can be found using the distance formula: \[ d = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} \] Substituting the coordinates:
- City A: \( (300, 0) \)
- City C: \( (-167.734, 103.008) \)
Thus, \[ d = \sqrt{(300 - (-167.734))^2 + (0 - 103.008)^2} \] \[ d = \sqrt{(300 + 167.734)^2 + (-103.008)^2} \] \[ d = \sqrt{(467.734)^2 + (-103.008)^2} \]
Calculating each component: \[ (467.734)^2 \approx 219,844.4 \] \[ (-103.008)^2 \approx 10,609.65 \]
Adding these: \[ d = \sqrt{219,844.4 + 10,609.65} \approx \sqrt{230,454.05} \approx 480.2 \text{ km} \]
So, the distance from City C to City A is approximately 480.2 km.
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