To determine the resultant speed and direction of the yacht, we can use vector addition.
1. Convert the given speed and direction into vector form. The yacht's speed due south can be represented as a vector (0, -12) knots.
2. Convert the current speed and direction into vector form. The current speed of 4 knots from bearing 040 degrees can be represented as a vector using trigonometry:
x-component: 4 * cos(40) = 3.06 knots
y-component: 4 * sin(40) = 2.57 knots
So, the current vector is (3.06, 2.57) knots.
3. Add the two vectors together:
Resultant vector = Yacht vector + Current vector
= (0, -12) + (3.06, 2.57)
= (3.06, -9.43) knots
4. Calculate the magnitude of the resultant vector, which represents the resultant speed of the yacht:
Magnitude = sqrt(3.06^2 + (-9.43)^2) = 9.84 knots
5. Calculate the direction of the resultant vector:
Direction = atan(-9.43/3.06) = -73.2 degrees
However, since the bearing should be measured clockwise from the north, we need to convert the direction to a positive angle:
Direction = 360 - 73.2 = 286.8 degrees
Therefore, the resultant speed of the yacht is approximately 9.84 knots, and the direction is approximately 286.8 degrees.
So, your answer of 15.28 knots from bearing 9.69 degrees is not correct. The correct answer is 9.84 knots from bearing 286.8 degrees.