A boat crosses a river and arrives at a point on the opposite bank directly across from its starting position. The boat travels at 4 m/s and the current is 2m/s. If the river is 600 m wide in what direction must the boat steer and how long will it take to cross?

want to go East

current 2 south
C = true compass heading
Vnorth = 0 = 4 cos C - 2
cos C = 1/2
C = 60 degrees East of North
Veast = 4 sin 60 = 4 (sqrt 3)/2 = 2 sqrt 3
600 = t * 2 sqrt 3
t = 300/sqrt 3

To determine the direction in which the boat must steer and how long it will take to cross the river, we can break down the problem into two components: the velocity of the boat relative to the ground (resultant velocity) and the angle at which the boat must travel.

Let's consider the velocity of the boat relative to the ground first. The velocity of the boat is given as 4 m/s, and the velocity of the river's current is 2 m/s. We can calculate the resultant velocity by using vector addition.

The resultant velocity (v) can be found using the Pythagorean theorem:

v = √(v^2_b + v^2_c)

Where v_b is the velocity of the boat and v_c is the velocity of the current.

v = √(4^2 + 2^2)
v = √(16 + 4)
v = √20
v ≈ 4.47 m/s

So, the resultant velocity of the boat relative to the ground is approximately 4.47 m/s.

Now, let's determine the angle at which the boat must travel to cross the river. We can use trigonometry to find the angle.

The angle (θ) can be found using the tangent function:

tan(θ) = v_c / v_b

Where v_c is the velocity of the current and v_b is the velocity of the boat.

tan(θ) = 2 / 4
θ = tan^(-1)(0.5)
θ ≈ 26.57 degrees

Therefore, the boat must steer at an angle of approximately 26.57 degrees with respect to the direction perpendicular to the riverbank.

Next, let's calculate the time it takes for the boat to cross the river. We can use the formula:

Time = Distance / Velocity

The distance to be covered by the boat is given as 600 m.

Time = 600 m / 4.47 m/s
Time ≈ 134.23 seconds (or approximately 2 minutes and 14.23 seconds)

So, the boat must steer at an angle of approximately 26.57 degrees and it will take approximately 2 minutes and 14.23 seconds to cross the river.