A boat heads 15 degrees west of north with a water speed of 3m/s. Determine its velocity relative to the ground when there is a 2m/s current from 40 degrees east of north. How long does it take to cross the river which is 800 m wide? Where does the boat land on the other side?

To find the velocity of the boat relative to the ground, we can use vector addition.

First, let's break down the boat's velocity and the current's velocity into their horizontal and vertical components.

The boat's velocity relative to the ground can be broken down as follows:
V(boat) = 3 m/s at an angle -15° west of north

The current's velocity can be broken down as follows:
V(current) = 2 m/s at an angle 40° east of north

Now, we need to find the resultant velocity (V(resultant)) by adding the horizontal and vertical components of the boat's velocity and the current's velocity.

The horizontal components can be calculated using trigonometry as follows:
H(boat) = V(boat) * sin(theta(boat))
H(current) = V(current) * sin(theta(current))

The vertical components can be calculated using trigonometry as follows:
V(boat) = V(boat) * cos(theta(boat))
V(current) = V(current) * cos(theta(current))

Now, we can calculate the horizontal and vertical components of the resultant velocity:
H(resultant) = H(boat) + H(current)
V(resultant) = V(boat) + V(current)

By substituting the given values, we can calculate the horizontal and vertical components of the resultant velocity.

Next, we can find the magnitude and direction of the resultant velocity using Pythagoras' theorem and trigonometry:
Magnitude of the resultant velocity: sqrt((H(resultant))^2 + (V(resultant))^2)
Direction of the resultant velocity: arctan(V(resultant) / H(resultant))

Substituting the calculated values, we can find the magnitude and direction of the resultant velocity.

Now, let's move on to calculating the time it takes for the boat to cross the river.

The time is given by the equation: t = d / V(resultant)
where d is the width of the river.

By substituting the given value of the width of the river and the calculated value of the resultant velocity, we can find the time taken by the boat to cross the river.

Lastly, to find where the boat lands on the other side of the river, we can use the horizontal component of the resultant velocity and the time it took for the boat to cross the river.

The horizontal distance traveled can be found by using the equation: distance = H(resultant) * t

By substituting the calculated values of the horizontal component of the resultant velocity and the time taken, we can find where the boat lands on the other side of the river.