A rowboat crosses a river with a velocity of 3.55 mi/h at an angle 62.5o north of west relative to the water. The river is 0.850 mi wide and carries an eastward current of 0.57 mi/h. How far upstream is the boat when it reaches the opposite shore?
To find how far upstream the boat is when it reaches the opposite shore, we can break down the given information into its components.
Let's start by considering the velocity of the boat relative to the ground. We have the velocity of the boat relative to the water, which is given as 3.55 mi/h at an angle of 62.5 degrees north of west. To find the velocity of the boat relative to the ground, we need to account for the current.
The eastward current has a velocity of 0.57 mi/h. To find the total velocity of the boat relative to the ground, we can use the following method:
1. Resolve the velocity of the boat relative to the water into its eastward and northward components:
- Eastward component (x-axis): 3.55 mi/h * cos(62.5 degrees)
- Northward component (y-axis): 3.55 mi/h * sin(62.5 degrees)
2. Add the eastward current's velocity to the eastward component of the boat's velocity:
- Total eastward velocity = eastward component of the boat's velocity + eastward current's velocity
3. The net westward velocity (opposite direction of eastward velocity) of the boat relative to the ground can be calculated as follows:
- Net westward velocity = total eastward velocity - eastward current's velocity
Now that we have the net westward velocity of the boat relative to the ground, we can find the time it takes for the boat to cross the river. The time can be calculated using the formula:
Time = Distance / Velocity
We already know the width of the river is 0.850 mi. So, the distance the boat crosses the river is 0.850 mi. Divide this distance by the net westward velocity of the boat to find the time.
Finally, since the boat travels at a constant velocity upstream (westward), we can calculate the distance it covers in that time by multiplying the time by the velocity of the boat relative to the ground.
Now, we have all the information needed to find how far upstream the boat is when it reaches the opposite shore.