Draw a diagram to scale showing the direction in which a man must row across a river in order to reach a point directly opposite, if he rows 3 mile/hr while the speed of current is 2 mile/hr.

My ans- with angle of 33' with straight line he wants to travel. And V=3.6 mile/hr.

Now if the river is 1/4 mile broad, how long does it take to cross it and what is the velocity of the boat relative to the shore.

Is this correct?
ans- 3.6 mile in 1 hr then for 0.25 mile t=4.16 min. And relative velocity is 3.6 mile/hr.

3 sin thete = 2

theta = 41.81

V across = 3 cos theta
= 2.24 mi/hr

(1/4)/2.24 = .112 hr = 6.7 minutes

He has zero velocity parallel to the shore (that is the whole point)
However he has speed of 2.24 mi/hr perpendicular to the shore.

To draw the diagram, start by drawing a straight line representing the direction the man wants to travel. Label it as his intended path. Then, draw a line angling downwards from the intended path to represent the direction of the current. Label it as the current. Next, draw a line above the intended path at a slightly bigger angle to represent the actual path that the man needs to take in order to counteract the current and reach the point directly opposite. Label this line as the man's path.

To determine the angle of the man's path, you can use the concept of vector addition. Since the man's velocity is 3 mph and the current's velocity is 2 mph, you can think of their velocities as vectors. The man's velocity vector points towards his intended path, while the current's velocity vector points in the opposite direction. To counteract the effect of the current, the man's path must be the resultant vector of the two velocities. You can use trigonometry to find the magnitude and direction of this resultant vector.

In this case, the angle of the man's path can be found using the tangent function. Since the current is 2 mph and the man's velocity is 3 mph, the tangent of the angle can be calculated as (2/3). Taking the inverse tangent of this value will give you the angle.

To determine how long it takes to cross the river, you can calculate the time using the formula d = vt, where d is the distance, v is the velocity, and t is the time. In this case, the distance is 0.25 miles and the velocity is 3.6 mph (which is the man's velocity relative to the shore). Solving for t, you would get t = d/v = 0.25/3.6 = 4.16 minutes.

The velocity of the boat relative to the shore is still 3.6 mph. This is because the velocity of the current does not affect the boat's velocity relative to the shore. The current only affects the direction in which the boat needs to travel to reach the point directly opposite.

Your answer is correct!