Final answer:
To take the shortest path southward across a river flowing west to east, a man should swim in a direction slightly west of south to compensate for the river's eastward current. This requires calculating the correct angle using vector addition.
Step-by-step explanation:
A man attempting to swim directly southward across a river that flows from west to east must contend with the river's current. To ensure the shortest possible path (which is directly south in this case), the man should aim his swim in a direction slightly west of south. This will compensate for the eastward flow of the river.
As an example, if a river flows from southwest to northeast at a velocity of 7.1 m/s, and a boat with a velocity of 13 m/s must reach a point due east, it would need to head in a direction to counteract the river flow. If we calculate this, the correct answer could be, for instance, heading 22.6° east of south.
To calculate the direction a man should aim to take the shortest path, one would need to perform a vector addition of the swimmer's velocity and the river's velocity. The swimmer must swim upstream at an angle that allows the resultant vector to be directly southward.