To calculate the distance covered from the starting point after the boat has sailed on the given bearings, we can break the problem down using vector components.
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Convert Bearings to Cartesian Coordinates:
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For the first leg (30 km at a bearing of 145°):
- The angle measured counter-clockwise from the positive x-axis (East) is \(145°\). Therefore, it can be represented in Cartesian coordinates as: \[ x_1 = 30 \cdot \cos(145°) \] \[ y_1 = 30 \cdot \sin(145°) \]
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For the second leg (10 km at a bearing of 250°):
- The angle measured counter-clockwise from the positive x-axis (East) is \(250°\). Therefore, it can be represented in Cartesian coordinates as: \[ x_2 = 10 \cdot \cos(250°) \] \[ y_2 = 10 \cdot \sin(250°) \]
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Calculate the Cartesian Coordinates:
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First leg: \[ x_1 = 30 \cdot \cos(145°) \approx 30 \cdot (-0.819) \approx -24.57 \text{ km} \] \[ y_1 = 30 \cdot \sin(145°) \approx 30 \cdot 0.5736 \approx 17.21 \text{ km} \]
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Second leg: \[ x_2 = 10 \cdot \cos(250°) \approx 10 \cdot (-0.17365) \approx -1.7365 \text{ km} \] \[ y_2 = 10 \cdot \sin(250°) \approx 10 \cdot (-0.9848) \approx -9.848 \text{ km} \]
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Add the Vectors:
- Total x-coordinate: \[ x_{\text{total}} = x_1 + x_2 \approx -24.57 - 1.7365 \approx -26.3065 \text{ km} \]
- Total y-coordinate: \[ y_{\text{total}} = y_1 + y_2 \approx 17.21 - 9.848 \approx 7.362 \text{ km} \]
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Calculate the Distance from the Starting Point:
- The distance from the starting point can be found using the Pythagorean theorem: \[ d = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2} \] \[ d = \sqrt{(-26.3065)^2 + (7.362)^2} \] \[ d = \sqrt{692.9992 + 54.1865} \approx \sqrt{747.1857} \approx 27.37 \text{ km} \]
Thus, the distance covered from the starting point is approximately 27.37 km.