a ferry boat is sailing at 12 km/h 30 degrees west of north with respect to a river that is flowing at 6 km/h. in what direction, as observe from the shore, is the ferry boat is sailing?
Due north
To determine the direction of the ferry boat as observed from the shore, we need to find the resultant velocity vector. We can break down the velocities into their respective components and then find their resultant.
Given:
Speed of the ferry boat (V_fb) = 12 km/h
Angle west of north (θ) = 30 degrees
Speed of the river (V_r) = 6 km/h
Using trigonometry, we can determine the component of the ferry boat's velocity in the north direction and west direction.
North component of the ferry boat's velocity (Vn_fb):
Vn_fb = V_fb * cos(θ)
= 12 * cos(30)
≈ 10.39 km/h
West component of the ferry boat's velocity (Vw_fb):
Vw_fb = V_fb * sin(θ)
= 12 * sin(30)
≈ 6 km/h
Since the river is flowing directly south, we can subtract its velocity from the northward component of the ferry boat's velocity:
Resultant velocity in the north direction (Vn_resultant):
Vn_resultant = Vn_fb - V_r
= 10.39 - 6
≈ 4.39 km/h north
The resultant velocity in the west direction (Vw_resultant) remains the same:
Vw_resultant = Vw_fb
= 6 km/h west
Now, we can find the direction as observed from the shore using the tangent function:
tan(θ_resultant) = Vw_resultant / Vn_resultant
θ_resultant = tan^(-1)(Vw_resultant / Vn_resultant)
= tan^(-1)(6 / 4.39)
≈ 53.4 degrees
Therefore, as observed from the shore, the ferry boat is sailing approximately 53.4 degrees west of north.
To determine the direction of the ferry boat as observed from the shore, we need to find the resultant velocity by combining the velocity of the boat with the velocity of the river.
The given information tells us that the ferry boat is sailing at 12 km/h at 30 degrees west of north, and the river is flowing at 6 km/h. Since the boat's velocity is specified relative to north, we will start by breaking down the given information into its northward and westward components.
The northward component of the boat's velocity can be found using trigonometry:
Northward velocity = Speed of the boat * sin(angle)
= 12 km/h * sin(30 degrees)
= 6 km/h * 0.5
= 3 km/h
The westward component of the boat's velocity can also be determined using trigonometry:
Westward velocity = Speed of the boat * cos(angle)
= 12 km/h * cos(30 degrees)
= 6 km/h * (√3/2)
= 6 km/h * 0.866
= 5.196 km/h
Now, let's combine the northward component of the boat's velocity with the river's velocity to find the resultant velocity:
Resultant northward velocity = Northward velocity of the boat + Northward velocity of the river
= 3 km/h + 0 km/h (since the river is flowing east-west, it has no northward component)
= 3 km/h
Resultant westward velocity = Westward velocity of the boat + Westward velocity of the river
= 5.196 km/h + 6 km/h
= 11.196 km/h
Finally, we can find the direction of the resultant velocity using trigonometry:
Angle = arctan(Resultant westward velocity / Resultant northward velocity)
= arctan(11.196 km/h / 3 km/h)
= arctan(3.732)
The inverse tangent of 3.732 is approximately 74.9 degrees.
So, as observed from the shore, the ferry boat is sailing at approximately 74.9 degrees east of north.