A) To find the magnitude of the resultant acceleration, we need to use the forces acting on the sailboat and divide it by the mass of the boat.
The eastward force due to the tide is 22000 N, which we can consider as positive. The northwest force due to the wind is 80900 N. To find the eastward component of the wind force, we can use the cosine of the angle between the northwest direction and the eastward direction, which is 45°. The eastward component of the wind force is 80900 N * cos(45°) = 80900 N * (√2/2) = 80900 N * 0.707 ≈ 57194 N.
Now we can calculate the resultant force in the east-west direction:
Resultant force = Eastward force + Eastward component of the wind force
= 22000 N + 57194 N
= 79194 N
To find the acceleration, we divide the resultant force by the mass of the sailboat:
Acceleration = Resultant force / Mass
= 79194 N / 15700 kg
≈ 5.04 m/s^2
Therefore, the magnitude of the resultant acceleration of the sailboat is approximately 5.04 m/s^2.
B) To find the direction of the boat's acceleration, we can use the vector sum of the forces. We have an eastward force of 22000 N and a northwest force of 80900 N.
To find the northward component of the wind force, we can use the sine of the angle between the northwest direction and the northward direction, which is 45°. The northward component of the wind force is 80900 N * sin(45°) = 80900 N * (√2/2) = 80900 N * 0.707 ≈ 57194 N.
Therefore, the northward force on the boat is equal to the northward component of the wind force, which is approximately 57194 N.
Since the eastward force due to the tide is much smaller than the wind force, the boat's acceleration will be in the northwest direction, close to the direction of the wind force.
Hence, the direction of the boat's acceleration is approximately 45° northward of West.