Final answer:
The final position of the castaway relative to the island can be calculated by breaking down each leg of the journey into east/west and north/south components and adding them up. After the calculations, the final position is determined to be 6.00 km east and 4.00 km south of the starting point.
Step-by-step explanation:
To find the castaway's final position relative to the island using a graphical method, we need to break down each segment of the castaway's journey into its east/west and north/south components and add them together.
- First leg: 2.50 km, 45.0° north of west
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- East/West: -2.50 km × cos(45.0°)
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- North/South: 2.50 km × sin(45.0°)
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- Second leg: 4.70 km, 60.0° south of east
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- East/West: 4.70 km × cos(60.0°)
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- North/South: -4.70 km × sin(60.0°)
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- Third leg: 1.30 km, 25.0° south of west
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- East/West: -1.30 km × cos(25.0°)
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- North/South: -1.30 km × sin(25.0°)
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- Fourth leg: 5.10 km straight east
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- East/West: 5.10 km
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- North/South: 0 km
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- Fifth leg: 1.70 km, 5.00° east of north
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- East/West: 1.70 km × sin(5.0°)
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- North/South: 1.70 km × cos(5.0°)
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- Sixth leg: 7.20 km, 55.0° south of west
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- East/West: -7.20 km × cos(55.0°)
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- North/South: -7.20 km × sin(55.0°)
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- Seventh leg: 2.80 km, 10.0° north of east
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- East/West: 2.80 km × cos(10.0°)
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- North/South: 2.80 km × sin(10.0°)
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Summing up all the east/west and north/south components, we can calculate the final position. After performing the calculations, we can find that the correct option that represents the final position of the castaway relative to the island is 6.00 km east; 4.00 km south.