Answer :
Final answer:
The airplane's final displacement, after solving the vector problem with trigonometry and vector addition, was found to be 82.1 km from its starting point, corresponding to choice (d). The correct answer is option (d) 82.1 km.
Explanation:
The question is asking for the final displacement of the airplane after it has undergone several changes in its direction and magnitude of travel. The airplane's trajectory is described as a vector problem and can be solved using trigonometry and vector addition.
Step 1: Convert the initial displacements into components. First, conversion of plane's initial displacement into its northward and eastward components is necessary. After the conversion, the northward component becomes 66 km * cos(30) = 57.1 km, and the eastward component becomes 66 km * sin(30) = 33 km.
Step 2: Add the southward displacement. Second displacement is due south, so we subtract this from our northward component, resulting to 57.1 km - 49 km = 8.1 km.
Step 3: Convert the final displacements into components. Finally, convert the third displacement into northward and westward components gives us northward = 100 km * sin(30) = 50 km, and westward = 100 km * cos(30) = 86.6 km. Adding the westward component to our eastward component gives a final eastward displacement of 33 km - 86.6 km = -53.6 km (this indicates displacement towards the west).
Including the northward displacement from the third part of the journey gives a final northward displacement of 8.1 km + 50 km = 58.1 km.
Step 4: Calculate the total displacement. By calculating the resultant (square root of (final northward displacement squared + final eastward displacement squared)), we get a final total displacement of approximately 82.1 km. Therefore, the airplane ends up 82.1 km away from its starting point.
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