The kayaker needs to paddle at an angle of 41.81º opposite to the current so that he can travel straight across the harbor, and it will take him 44.73 seconds to cross.
This is a 2-dimensional problem, since the kayaker can move in 2 dimensions over the river. For our study, we will model how does the kayaker move in the direction transversal to the river (meaning across the river), and longitudinal to the river (meaning along the river).
On the first part of this problem, we're asked to compute the direction at which the kayaker needs to paddle so that he moves only transversal to the river flow. Since the river flow has a velocity of 2 meters per second on the longitudinal direction, then the kayaker must have a velocity opposite to the river flow on the longitudinal direction of 2 meters per second.
Since he can paddle with a speed of 3 meters per second, we can make a right triangle (check the attached figure), and through trigonometry find the angle at which he needs to paddle against the river to only move in the transversal direction. Therefor we can write:
[tex]sin(\theta) = \frac{2}{3}[/tex]
Therefor [tex]\theta[/tex] is 41.81 degrees. This means that the kayaker moves across the river with a velocity 2.24 meters per second. With this velocity and the width of the river, we can easily find that the time it will take to cross is 44.73 seconds.
Right triangle, velocity, sine of angle.
