A hunter wishes to cross a river that is 1.1 km wide and flows with a speed of 5.0 km/h parallel to its banks. The hunter uses a small powerboat that moves at a maximum speed of 9 km/h with respect to the water. What is the minimum time necessary for crossing?

To determine the minimum time necessary for crossing, we need to consider the relative velocity of the boat with respect to the river.

Step 1: Calculate the velocity of the river (Vr)
Given that the river flows parallel to its banks with a speed of 5.0 km/h, the velocity of the river is 5.0 km/h in the same direction.

Step 2: Calculate the velocity of the boat with respect to the ground (Vb)
The boat has a maximum speed of 9.0 km/h with respect to the water.

Step 3: Determine the angle between the direction of the boat's motion and the direction of the river flow (θ)
Since the boat needs to cross the river, it will need to steer at an angle with respect to the river flow. Let's assume that angle is θ.

Step 4: Use vector addition to find the resultant velocity (Vr) of the boat with respect to the ground
To find the resultant velocity (Vr) of the boat with respect to the ground, we need to break down the velocity of the boat into its components. The horizontal component is Vb cos(θ), and the vertical component is Vb sin(θ). Since the boat needs to cross the river, the horizontal component (Vb cos(θ)) should be equal to the velocity of the river (Vr).

Vb cos(θ) = Vr
9.0 km/h * cos(θ) = 5.0 km/h
cos(θ) = 5.0 km/h / 9.0 km/h
cos(θ) = 0.556

Taking the inverse cosine of 0.556, we can find the angle θ:
θ = cos^(-1)(0.556) = 55.4 degrees

Step 5: Calculate the time required to cross the river (t)
We can use the known distance of the river (1.1 km) and the horizontal component of the boat's velocity (Vb cos(θ)) to find the time required to cross the river (t).

t = distance / velocity
t = 1.1 km / (9.0 km/h * cos(θ))
t = 1.1 km / (9.0 km/h * 0.556)
t = 0.134 hours

Step 6: Convert the time to minutes or seconds (optional)
If you wish to convert the time to minutes or seconds, you can multiply the decimal portion of the time (0.134) by 60 to convert it to minutes, or by 3600 to convert it to seconds.

In this case, the minimum time necessary for crossing is approximately 0.134 hours, which is equivalent to approximately 8.04 minutes or 482.4 seconds.