A rowboat crosses a river with a velocity of 3.42 mi/h at an angle 62.5° north of west relative to the water. The river is 0.610 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 determine the boat's actual velocity.
First, we can start by decomposing the boat's velocity into horizontal and vertical components. Since the boat is moving at an angle north of west, we can use trigonometry to find the horizontal and vertical components.
Horizontal component = velocity * cos(angle)
Vertical component = velocity * sin(angle)
Horizontal component = 3.42 mi/h * cos(62.5°) ≈ 1.54 mi/h
Vertical component = 3.42 mi/h * sin(62.5°) ≈ 2.90 mi/h
Now, let's consider the effect of the river's current. The current has a velocity of 1.25 mi/h eastward, which means it impacts the boat's overall velocity.
In the horizontal direction, the current and boat's horizontal components add up:
Horizontal velocity = current velocity + horizontal boat velocity
Horizontal velocity = 1.25 mi/h + 1.54 mi/h ≈ 2.79 mi/h
In the vertical direction, the boat's vertical component remains unchanged: 2.90 mi/h.
Now we can calculate the time it takes for the boat to cross the river, using the river's width as the distance:
Time = distance / velocity
Time = 0.610 mi / 2.79 mi/h ≈ 0.219 hours
Finally, we need to find the distance the boat has traveled upstream during this time.
Distance upstream = vertical component * time
Distance upstream = 2.90 mi/h * 0.219 hours ≈ 0.634 mi
Therefore, the boat is approximately 0.634 miles upstream when it reaches the opposite shore.