To find the distance between the man's starting point and his final position, we can use vector addition.
Let's call the man's starting point A and his final position B. We can represent the two legs of his journey as vectors A and B, respectively.
The first leg of his journey, due west, can be represented as a vector of magnitude 4 km in the -x direction.
The second leg of his journey, on a bearing of 197°, can be represented as a vector in the -x and -y direction. We can use trigonometry to find the components of this vector.
The x component of the vector can be found using the formula:
x = magnitude * cos(angle) = d * cos(197°)
The y component of the vector can be found using the formula:
y = magnitude * sin(angle) = d * sin(197°)
where d is the magnitude of the vector (the distance the man traveled in this leg of his journey).
To find the magnitude of the vector, we'll use the Pythagorean theorem:
d^2 = x^2 + y^2
We can substitute the x and y components we found above into this equation to find d:
d^2 = (d * cos(197°))^2 + (d * sin(197°))^2
d^2 = d^2 * (cos^2(197°) + sin^2(197°))
d^2 = d^2
d = sqrt(d^2) = sqrt(d^2)
So the magnitude of the vector is d.
Next, we can add the two vectors to find the total distance from the man's starting point to his final position:
distance = sqrt((4 - d * cos(197°))^2 + (d * sin(197°))^2)
This is the distance the man is from his starting point. To find the exact value, we would need to know the value of d. However, without that information, this is as far as we can take the calculation.