Final answer:
To find the true speed and direction of the motorboat, one must vectorially add the river's velocity to the motorboat's velocity, resolve into components, and calculate the resultant magnitude and angle using trigonometric methods.
Step-by-step explanation:
The true speed and direction of the motorboat can be determined by using vector addition to combine the velocity of the motorboat relative to the water and the velocity of the river. Thus, we construct a right-angled triangle where the motorboat's speed relative to the water (12 mi/h) is the hypotenuse, and one side is the river's velocity (8 mi/h) heading due east. The angle between the motorboat's velocity and the due east direction is 30°.
Step 1: Resolve the Motorboat's Velocity
Resolve the motorboat's velocity into two components: One parallel to the river's flow (Vx) and one perpendicular to the river (Vy).
Vx = 12 mi/h * cos(30°)
Vy = 12 mi/h * sin(30°)
Step 2: Add Velocities
Combine the river velocity with the x-component of the motorboat's velocity to find the true eastward velocity relative to the shore (Vt,x).
Vt,x = Vx + river velocity
Step 3: Calculate the True Speed of the Motorboat
The true speed (Vt) is the resultant velocity magnitude, which can be found using the Pythagorean theorem.
Vt = √(Vt,x² + Vy²)
Step 4: Determine the True Direction
To find the angle (θ) of the true direction, use the tangent ratio since we have opposite (Vy) and adjacent (Vt,x) sides.
θ = arctan(Vy / Vt,x)