A kayaker paddles across a river at a constant speed of 4 miles per hour and a bearing of N 60 degrees E. Which current vector results in the kayaker's actual speed being 5.9 miles per hour and the kayakers actual direction being N 70 degrees E.

To find the current vector that results in the kayaker's actual speed being 5.9 miles per hour and the kayaker's actual direction being N 70 degrees E, we can use vector addition.

Let's call the current vector C, which we need to find.

The kayaker's velocity vector V can be broken down into two components, one parallel to the current vector and one perpendicular to it. The component parallel to the current vector will contribute to the kayaker's actual speed, while the perpendicular component will only affect the direction.

We can find the magnitude of the parallel component using the equation:

V_parallel = V_actual_speed - C_magnitude

Substituting the given values:

V_parallel = 5.9 mph - C_magnitude

Next, we can find the magnitude of the perpendicular component using the equation:

V_perpendicular = C_magnitude * tan(angle_difference)

The angle difference is the absolute difference between the given kayaker's direction (N 70 degrees E) and the bearing (N 60 degrees E):

angle_difference = |70 - 60| = 10 degrees

Substituting the given values:

V_perpendicular = C_magnitude * tan(10 degrees)

Now, we can use the Pythagorean theorem to find the magnitude of the kayaker's velocity vector:

V_magnitude = sqrt(V_parallel^2 + V_perpendicular^2)

Substituting the earlier equations:

5.9 mph = sqrt((5.9 mph - C_magnitude)^2 + (C_magnitude * tan(10 degrees))^2)

Solving this equation will give us the magnitude of the current vector, C_magnitude. Let's solve it step-by-step.

1. Square both sides of the equation:

34.81 mph^2 = (5.9 mph - C_magnitude)^2 + (C_magnitude * tan(10 degrees))^2

2. Expand the equation:

34.81 mph^2 = 34.81 mph^2 - 11.8 mph * C_magnitude + C_magnitude^2 + (C_magnitude * tan(10 degrees))^2

3. Simplify:

0 = C_magnitude^2 - 11.8 mph * C_magnitude + (C_magnitude * tan(10 degrees))^2

4. Use the quadratic formula:

C_magnitude = [-(b) ± sqrt(b^2 - 4ac)] / (2a)

In this case, a = 1, b = -11.8 mph, and c = (C_magnitude * tan(10 degrees))^2.

Solving this quadratic equation for C_magnitude will give you two solutions. One of them will be negative, which does not make sense in this context, so we can discard it.

By solving the equation using the quadratic formula, we can find the magnitude of the current vector, C_magnitude.

To find the current vector that results in the kayaker's actual speed and direction, we need to determine the difference between the desired and actual vector of the kayaker.

Let's break down the given information:
- The kayaker's desired speed is 5.9 miles per hour. This is the resultant speed of the kayaker and the current vector.
- The kayaker's desired direction is N 70 degrees E. This is the resultant direction of the kayaker and the current vector.

Next, we break down the kayaker's original speed and direction to determine the difference:
- The kayaker's original speed is 4 miles per hour.
- The kayaker's original direction is N 60 degrees E.

To find the difference between the desired and original vectors, we can subtract the original vector from the desired vector.

Taking into account the speed and direction components, we subtract the original speed and direction components from the desired speed and direction components:

Desired speed component (north): 5.9 * sin(70) = 5.83 miles per hour
Desired speed component (east): 5.9 * cos(70) = 2.04 miles per hour

Original speed component (north): 4 * sin(60) = 3.46 miles per hour
Original speed component (east): 4 * cos(60) = 2 miles per hour

The difference in the north component can be obtained by subtracting the original north component from the desired north component:
Difference in the north component = Desired speed component (north) - Original speed component (north) = 5.83 - 3.46 = 2.37 miles per hour

The difference in the east component can be obtained by subtracting the original east component from the desired east component:
Difference in the east component = Desired speed component (east) - Original speed component (east) = 2.04 - 2 = 0.04 miles per hour

Therefore, the current vector that results in the kayaker's actual speed being 5.9 miles per hour and their actual direction being N 70 degrees E is:
Magnitude: sqrt((Difference in north component)^2 + (Difference in east component)^2) = sqrt((2.37)^2 + (0.04)^2)
Direction: arctan(Difference in east component / Difference in north component)

After calculating, we find that the magnitude of the current vector is approximately 2.38 miles per hour, and the direction is approximately N 1 degree E.

Hence, the current vector that produces the desired speed and direction for the kayaker is approximately 2.38 miles per hour at a bearing of N 1 degree E.

If we let

u = 4.0 @ N60E = (3.464,2.000)
w = 5.9 @ N70E = (5.544,2.018)

we want v such that

u+v = w

That should help.