At the beginning of a race, a yacht crosses the start line and sails in a westerly direction at a speed of 6 m s-1. The current of the sea is flowing at 2 m s-1 towards the north-east.

Calculate how far and in what direction (relative to west)the yacht has travelled after 18 minutes from its crossing the start line.

After you make your sketch you could use the cosine law

x^2 = 2^2 + 6^2 - 2(2)(6)cos 45
...
x = 4.799
then by the sine law,
sin theta/2 = sin45/4.799
theta = appr 17.14 degrees

or using vectors
resultant = (6cos180,6sin180) + (2cos45,2sin45)
= (-6,0) + (1.4142, 1.4142)
= (-4.5858,1.4142)
magnitude = sqrt(-4.5858^2 + 1.4142^2)
= 4.799 , same as above

for angle:
tan theta = 1.4142/4.5858
theta = 17.139 same a before

state the proper conclusion using necessary units

To calculate the distance and direction of the yacht after 18 minutes, we need to break down the velocity vectors of the yacht and the current into their respective components.

Given:
Yacht velocity (v_yacht) = 6 m/s towards the west
Current velocity (v_current) = 2 m/s towards the north-east

Step 1: Convert time to seconds
18 minutes = 18 * 60 = 1080 seconds

Step 2: Calculate the displacement of the yacht after 18 minutes
Displacement of the yacht (d_yacht) = v_yacht * t
d_yacht = 6 m/s * 1080 s = 6480 m towards the west

Step 3: Calculate the displacement caused by the current after 18 minutes
Since the current is flowing towards the north-east, it can be broken down into its north and east components.

The north component of the current (v_current_north) = v_current * sin(45°) = 2 m/s * sin(45°) = 2 * 0.7071 = 1.4142 m/s towards the north
The east component of the current (v_current_east) = v_current * cos(45°) = 2 m/s * cos(45°) = 2 * 0.7071 = 1.4142 m/s towards the east

Displacement caused by the current (d_current) = v_current_north * t + v_current_east * t
d_current = 1.4142 m/s * 1080 s + 1.4142 m/s * 1080 s
d_current = 1527.88 m towards the north + 1527.88 m towards the east

Step 4: Find the resultant displacement
To find the resultant displacement, combine the displacement of the yacht (d_yacht) and the displacement caused by the current (d_current) using vector addition.

Resultant displacement = d_yacht + d_current
Resultant displacement = 6480 m towards the west + 1527.88 m towards the north + 1527.88 m towards the east

Step 5: Calculate the direction
To find the direction, we can use trigonometry to calculate the angle relative to west. We will use the arctan function.

Direction (θ) = arctan(d_current_east / d_current_west)
θ = arctan(1527.88 m / 6480 m)

Calculate the angle using a scientific calculator to get the result.

So, after 18 minutes, the yacht has traveled approximately 6480 meters towards the west, 1527.88 meters towards the north, and 1527.88 meters towards the east. The direction (relative to west) will be calculated using the arctan function.

To calculate the distance and direction the yacht has traveled after 18 minutes, we need to break down the yacht's velocity into its components and determine how far it has traveled in both the west and north directions.

First, let's convert the time from minutes to seconds:
18 minutes = 18 * 60 = 1080 seconds.

Now, we can calculate the distance the yacht has traveled in the west direction (x-component) and in the north direction (y-component) separately.

Distance traveled in the west direction:
The yacht's velocity in the west direction is given as 6 m/s. Since the yacht is moving at a constant speed and in a straight line, we can use the formula: distance = speed * time.

Distance west = velocity west * time = (6 m/s) * (1080 s) = 6480 meters west.

Distance traveled in the north direction:
The current is flowing at 2 m/s towards the north-east. To determine the northward component of the yacht's velocity, we need to find the northward component of the current velocity.

The northward component can be found by multiplying the current velocity by the sine of the angle it makes with the north direction. Since the current is flowing towards the north-east, it makes a 45-degree angle with the north direction.

Northward component = current velocity * sin(45°) = (2 m/s) * sin(45°) ≈ 1.41 m/s.

Now, we can calculate the distance traveled in the north direction using the yacht's velocity and the current's northward component:

Distance north = (velocity north + current's northward component) * time
= (0 m/s + 1.41 m/s) * (1080 s) = 1522.8 meters north.

To find the total distance traveled by the yacht, we can use the Pythagorean theorem, which states that the sum of the squares of the two perpendicular sides of a right triangle is equal to the square of the hypotenuse.

Total distance = sqrt((distance west)^2 + (distance north)^2)
= sqrt((6480 m)^2 + (1522.8 m)^2)
≈ 6735.4 meters.

Finally, we can find the direction the yacht has traveled relative to west by calculating the angle of the total distance with respect to the west direction. This angle can be found using the inverse tangent function:

Angle = atan(distance north / distance west)
= atan(1522.8 m / 6480 m)
≈ 13.47°.

Therefore, after 18 minutes, the yacht has traveled approximately 6735.4 meters at an angle of about 13.47° relative to the west direction.