This problem involves the concepts of vectors and trigonometry. A vector can be broken down into two perpendicular components which can be determined using trigonometric principles.
First, the velocity vector for the current flowing east is designated as (0, 2.5 m/s).
Next, the swimmer's velocity, which we'll say is directed northeast for now, can be represented as (vsin(theta), vcos(theta)), where "v" represents the magnitude of the velocity (8.5 m/s) and "theta" is the angle between the velocity vector and the north.
Our goal is to find the value of theta so that the swimmer ends up directly across on the other shore (north).
Starting with the Pythagorean theorem, the eastward component of the swimmer's velocity, vsin(theta), must equal the eastward velocity of the river's current, while the northward component of the swimmer's velocity, vcos(theta), ensures that the swimmer travels north.
By equating vsin(theta) to the river's current, we get:
vsin(theta) = 2.5
Rearrange the equation to solve for theta:
sin(theta) = 2.5/v
Plug in the velocity of the swimmer:
sin(theta) = 2.5/8.5
Solve for theta by taking the inverse sine (asin or arcsin) of both sides.
theta = arcsin(2.5/8.5) This is our solution in radians, which roughly equals 0.2985 radians.
However, angle measurements are more commonly given in degrees in everyday contexts. We can use the fact that pi radians is equal to 180 degrees to convert our answer to degrees.
theta = degrees(0.2985 radians) = 17.10 degrees
So, the swimmer should aim roughly 17.10 degrees east of directly across the river to reach the point on the north shore directly opposite his/her starting point.