Final answer:
The man must row in a northwest direction with a resultant velocity of 4.47 km/hr to reach the opposite shore directly across the river. It will take approximately 1 hour to cross the river, which is 5 km wide.
Step-by-step explanation:
To determine the direction in which the man must set off to reach a position exactly opposite his starting point on the other side of a river, we need to consider the velocities of the man and the river. The man can row at 6 km/hr in still water, and the river's flow is 4 km/hr eastward. To offset the eastward flow of the river, the man must row in a northwest direction to ensure he reaches the point directly opposite.
Now, let's calculate the time it will take to cross the 5 km width of the river. Since the flow of the river is perpendicular to the desired crossing path, we must use the Pythagorean theorem to find the resultant velocity at which the man will actually be crossing. The resultant velocity (Vr) can be calculated as follows:
- Vr = sqrt( Vm² - Vr² )
- Vr = sqrt( 6² - 4² )
- Vr = sqrt( 36 - 16 )
- Vr = sqrt( 20 )
- Vr = 4.47 km/hr
Now calculating the time (t) to cross the river:
- t = distance / velocity
- t = 5 km / 4.47 km/hr
- t ≈ 1.12 hours
This value is closest to 1 hour, which aligns with the options provided in the question. Thus, the man must row in a northwest direction, and it will take approximately 1 hour to reach the opposite shore directly across the river.