Explanation:
To solve this problem, we can use the law of cosines. The law of cosines states that for any triangle with sides a, b, and c, and an angle C opposite side c,
c² = a² + b² - 2abcos(C)
In this problem, we can consider the man's starting point as the origin, and the distance he travels in the first leg as the first side of the triangle (a = 4km). The distance he travels in the second leg is the second side of the triangle (b). The angle between the two legs is the angle opposite side c, which is 90 degrees since the man changes direction from due west to southwest.
Therefore, we can use the law of cosines to find the distance he traveled in the second leg:
c² = a² + b² - 2abcos(90)
c² = 4² + b²
c = √(4² + b²)
c = √(16 + b²)
We know that the bearing of the second leg is 197 degrees, which is equivalent to a bearing of -163 degrees, since bearing measurements are taken clockwise from due north.
So we can use the law of cosines to find the distance he traveled in the second leg:
b = √(c² + a² - 2ac cos (-163))
We can now plug in the values we know:
b = √(c² + 4² - 24c*cos(-163))
We can now solve for c:
c = √(b² + 16 - 8bc)
The man is then southwest of his starting point, so the distance from his starting point to the final point is c.