City A lies 30 km directly south of city B. A bus, beginning at city A travels 50 km at 37° north of east to reach city C. How far and in what direction must the bus go from city C to reach city B?
A)20 km, west
B)40 km, west
C)80 km, west
D)40 km, east
E)80 km, east

Question

Quizplus · Accepted Answer

Using vector addition, we can draw a diagram to represent the distances and directions traveled by the bus:
     C
     /   
 50/  37°\
 /        \
A---------B
We can see that the bus has traveled 30 km south and 50 km at a 37° angle north of east, which can be broken down into horizontal and vertical components:
Horizontal: 50 cos(37°) = 39.6 km east
Vertical: 50 sin(37°) = 30 km north
Therefore, city C is 39.6 km east and 30 km north of city A. To get to city B, the bus must travel directly west, which is the opposite direction of the vector from C to A. Using the Pythagorean theorem, we can find the distance from C to B:
distance = sqrt((39.6 km)^2 + (30 km)^2) = 49.8 km
Therefore, the bus must go 49.8 km west from city C to reach city B, which is closest to choice B (40 km, west).

City a Lies 30 Km Directly South of City B