plz answer my question
a river is 2 km wide and flows at 4km/h. a motorboat has a speed of 10 km/h in still water and heads out from one bank. a marina is directly across the river, on the opposite bank. if the motorboat heads directly toward the marina, how far downstream from the marina will be on the other bank? USING VECTOR COMPONENTS
. It takes 2/10 hours to cross the river. In that time, the boat drifts downstream 2/10*4 = 4/5 km. is that right
Now, if i want to find the heading needed to get straight across the river, then i need some vectors. but how am i support to do it?
To find the heading needed to get straight across the river using vector components, you can break down the velocities of the boat and the river into their components.
Since the river is flowing at 4 km/h directly across, its velocity can be represented as a vector with horizontal component 4 km/h and vertical component 0 km/h.
The motorboat's speed in still water is 10 km/h. To break down its velocity, you need to consider the angle it makes with the river's flow. Let's call this angle theta.
The horizontal component of the motorboat's velocity is given by 10 km/h * cos(theta), and the vertical component is given by 10 km/h * sin(theta).
To head directly toward the marina, the boat's heading should be such that its vertical component cancels out with the river's vertical component, resulting in a net vertical velocity of 0 km/h. Therefore, we have:
10 km/h * sin(theta) - 4 km/h = 0
Simplifying the equation, we find:
10 km/h * sin(theta) = 4 km/h
Now, we can solve for theta by dividing both sides of the equation by 10 km/h:
sin(theta) = 4 km/h / 10 km/h
sin(theta) = 2/5
To find the angle theta, you can take the inverse sine (or arcsine) of both sides of the equation:
theta = arcsin(2/5)
Using a scientific calculator, you can find that theta is approximately 24.58 degrees.
Therefore, to head directly across the river, the boat needs to be angled at approximately 24.58 degrees upstream from directly across.
To calculate the distance downstream from the marina where the boat will be on the other bank, you were correct in your initial calculation. Multiplying the time it takes to cross the river (2/10 hours) by the speed of the river's flow (4 km/h), you get:
(2/10) hours * 4 km/h = 4/5 km
So, the boat will drift downstream approximately 4/5 km while crossing the river.