Final answer:
By using the law of cosines and the law of sines, the distance and bearing from the base camp can be determined after calculating the result of the two driving directions specified in the question.
Step-by-step explanation:
To find how far the surveyor is from the base camp and the bearing from it after the described movements, we can use the law of cosines and the law of sines from trigonometry. Since the movements can be represented as a vector diagram with two vectors at an angle to each other, these laws are applicable.
Firstly, the surveyor drives 42km on a bearing of 032, which means she travels northeast. Then she drives 28km on a bearing of 154, which means she travels southeast. Since the bearing is always measured from the north in a clockwise direction, the angle between the two paths is 154 - 32 = 122 degrees.
We then construct a triangle where the sides represent the paths traveled and the included angle is 122 degrees. To find the distance from the base camp, we apply the law of cosines:
d^2 = 42^2 + 28^2 - 2 × 42 × 28 × cos(122°)
After calculating the value of d, we determine the bearing using the law of sines to find the angle at the base camp and then convert this angle into a bearing.