Final answer:
The boat's velocity with respect to Earth is calculated by summing the vector components of the boat's and river's velocities. The resultant velocity is found to be 4.06 m/s, and the time to cross the river is 514.29 seconds. The provided answer options are incorrect.
Step-by-step explanation:
To solve this problem, we will need to consider the velocity of the river and the velocity of the boat independently and then combine them to find the resultant velocity, which is the velocity of the boat with respect to the Earth. The boat heads at 7.0 m/s at an angle of 30° north of west, while the river flows east at 4.0 m/s.
First, we break down the boat's velocity into components. The westward component of the boat's velocity (Vboat,x) is 7.0 m/s × cos(30°) = 6.06 m/s and the northward component (Vboat,y) is 7.0 m/s × sin(30°) = 3.5 m/s. Since the river is moving eastward and the boat is trying to move westward, their velocities in the east-west direction will oppose each other. Thus, the resultant horizontal velocity (Vtot,x) with respect to the Earth is Vboat,x - Vriver = 6.06 m/s - 4.0 m/s = 2.06 m/s westward.
The northward component remains unopposed, so the resultant northward velocity (Vtot,y) is simply 3.5 m/s. Now, we can calculate the magnitude of the boat's resultant velocity (Vtot) using the Pythagorean theorem: Vtot = √(Vtot,x² + Vtot,y²) = √(2.06² + 3.5²) m/s = √(4.2436 + 12.25) m/s = √16.4936 m/s = 4.06 m/s.
The time (t) it takes the boat to cross the river is the width of the river divided by the northward velocity component: t = width / Vboat,y = 1800 m / 3.5 m/s = 514.29 s.
In conclusion, the velocity of the boat with respect to Earth is 4.06 m/s, and the time it takes for the boat to cross the river is 514.29 s. None of the answer choices provided are correct.