A hunter wishes to cross a river that is 1.6 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 15 km/h with respect to the water. What is the minimum time necessary for crossing?
min
depends on whether he cares how far downstream the boat drifts. Minimum time is achieved by heading perpendicular to the bank. In that case, he has to go 1.6km at a speed of 15km/hr.
min time is thus 1.6km / 15 km/hr = .10666 hr = 6.4min
To find the minimum time necessary for crossing the river, we need to determine the angle at which the hunter should steer the boat across the river.
The river flows with a speed of 5.0 km/h parallel to its banks, which means the boat is moving against this current when crossing the river. This speed of the current affects the boat's effective speed.
Let's consider the scenario where the boat crosses the river directly perpendicular to the bank.
The boat's actual speed across the river (perpendicular to the bank) can be found using the Pythagorean theorem:
(actual speed across the river)^2 = (speed of the boat)^2 - (speed of the current)^2
(actual speed across the river)^2 = (15 km/h)^2 - (5 km/h)^2
(actual speed across the river)^2 = 225 km^2/h^2 - 25 km^2/h^2
(actual speed across the river)^2 = 200 km^2/h^2
actual speed across the river = sqrt(200) km/h
actual speed across the river ≈ 14.14 km/h
Now, we can calculate the time required to cross the river using the formula:
time = distance / speed
time = 1.6 km / 14.14 km/h
time ≈ 0.113 hours
Therefore, the minimum time necessary for crossing the river is approximately 0.113 hours, which is equivalent to 6.8 minutes.