A rowboat crosses a river with a velocity of 3.37 mi/h at an angle 62.5° north of west relative to the water. The river is 0.590 mi wide and carries an eastward current of 1.25 mi/h. How far upstream is the boat when it reaches the opposite shore?
To find out how far upstream the boat is when it reaches the opposite shore, we need to break down the motion of the boat into its horizontal and vertical components.
First, let's find the horizontal component of the boat's velocity. We can do this by multiplying the boat's speed (3.37 mi/h) by the cosine of the angle (62.5°):
Horizontal component = 3.37 mi/h * cos(62.5°)
Next, let's find the time it takes for the boat to cross the river. We can use the width of the river (0.590 mi) and the horizontal component of the boat's velocity to calculate this:
Time = Distance / Horizontal component
Now, since the boat is traveling upstream, against the current, the effective velocity of the boat will be the difference between its velocity and the current's velocity. So, the effective velocity of the boat is:
Effective velocity = Boat's velocity - Current's velocity
Finally, to find how far upstream the boat is, we multiply the effective velocity by the time it takes to cross the river:
Distance upstream = Effective velocity * Time
Let's plug in the given values and calculate the answer: