Final answer:
Using the analytical method to calculate vectors, the delivery driver's net displacement is 135 km, distance traveled is 170 km, return displacement is 135 km, and the compass heading is approximately 9.46° east of south.
Step-by-step explanation:
The delivery driver's trip consists of a series of vector displacements which can be analyzed using the analytical method to determine the net displacement vector, the distance to the restaurant, the return trip's displacement vector, and its compass heading. The trip includes driving 40 km north, 20 km west, 60 km northeast, and 50 km north.
To find the net displacement vector, we first break down the northeast vector into its north and east components by using trigonometric functions given that the direction northeast (NE) is 45° from both the north and east axis. Considering that northeast is equally between north and east, the north and east components of the 60 km NE displacement are both 60 km * cos(45°) = 42.42 km to the north and 42.42 km to the east. Now we sum up the north and east components of all legs of the trip:
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- North: 40 km + 42.42 km (NE component) + 50 km = 132.42 km north
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- East: 42.42 km (NE component) - 20 km = 22.42 km east
The net displacement vector can then be calculated by using the Pythagorean theorem: √(132.42² + 22.42²) = approx. 135 km. To find the direction from the post office to the restaurant, we take the arctan of the north component over the east component: arctan(132.42/22.42) = approx. 80.54° from the east, which means 9.46° west of north.
The distance to the restaurant is the total distance traveled, so we add up all distances: 40 km + 20 km + 60 km + 50 km = 170 km.
Since the displacement is a vector pointing from the post office to the restaurant, the return displacement vector will be the same magnitude but in the opposite direction (southwest). Therefore, the return displacement vector is 135 km, and the heading will be 180 - 9.46° = 170.54° or 9.46° east of south.
The correct option that matches the calculations is not provided in the multiple-choice answers, indicating a potential error in the question or the answer choices.