A rowboat crosses a river with a velocity of 3.30 mi/h at an angle 62.5° north of west relative to the water. The river is 0.795 mi wide and carries an eastward current of 1.25 mi/h. How far upstream is the boat when it reaches the opposite shore?

Well, that rowboat really knows how to navigate in style! Alright, let's get this boat party started!

First, we need to figure out the boat's velocity relative to the ground. To do that, we can break it down into its horizontal and vertical components. The vertical component can be found by multiplying the boat's velocity by the sine of the angle:

Vertical component = 3.30 mi/h * sin(62.5°)

Now, the horizontal component can be found using the cosine of the angle:

Horizontal component = 3.30 mi/h * cos(62.5°)

Now that we know the boat's velocity relative to the ground, we can find out how long it takes for the boat to cross the river. The time it takes for the boat to cross is given by:

time = distance / velocity

The distance the boat travels across the river is equal to the width of the river, which is 0.795 mi. The velocity of the boat relative to the ground is the horizontal component we calculated earlier. So now we can plug in these values:

time = 0.795 mi / (3.30 mi/h * cos(62.5°))

Once we know the time it takes for the boat to cross the river, we can find out how far upstream it is by multiplying the time by the upstream current. The time it takes for the boat to reach the opposite shore is the same time we calculated above. So the distance upstream is given by:

distance upstream = time * upstream current

distance upstream = (0.795 mi / (3.30 mi/h * cos(62.5°))) * 1.25 mi/h

After doing all these calculations, the boat will end up about 0.154 mi upstream when it reaches the opposite shore. That's not too shabby! So make sure to pack your water wings if you're on the same boat, my friend!

To find how far upstream the rowboat is when it reaches the opposite shore, we need to determine the boat's effective velocity and the time it takes to cross the river.

First, let's break down the boat's velocity into its components.

Given:
Boat velocity = 3.30 mi/h
Angle north of west = 62.5°
Current velocity = 1.25 mi/h
River width = 0.795 mi

1. Calculate the westward component of the boat's velocity:
V_west = Boat velocity * cos(Angle)
V_west = 3.30 mi/h * cos(62.5°)

2. Calculate the northward component of the boat's velocity:
V_north = Boat velocity * sin(Angle)
V_north = 3.30 mi/h * sin(62.5°)

3. Calculate the effective velocity of the boat across the river:
V_effective = sqrt(V_west^2 + (Current velocity)^2)
V_effective = sqrt(V_west^2 + (1.25 mi/h)^2)

4. Calculate the time it takes for the boat to cross the river:
Time = River width / V_effective
Time = 0.795 mi / V_effective

5. Calculate how far upstream the boat has traveled:
Distance upstream = Time * V_west

Let's calculate the values step-by-step:

1. Calculate V_west:
V_west = 3.30 mi/h * cos(62.5°)
V_west ≈ 1.410 mi/h

2. Calculate V_north:
V_north = 3.30 mi/h * sin(62.5°)
V_north ≈ 2.885 mi/h

3. Calculate V_effective:
V_effective = sqrt((1.410 mi/h)^2 + (1.25 mi/h)^2)
V_effective ≈ sqrt(1.987025 + 1.5625)
V_effective ≈ sqrt(3.549525)
V_effective ≈ 1.882 mi/h

4. Calculate the time:
Time = 0.795 mi / 1.882 mi/h
Time ≈ 0.422 hours

5. Calculate the distance upstream:
Distance upstream = 0.422 hours * 1.410 mi/h
Distance upstream ≈ 0.595 mi

Therefore, the boat is approximately 0.595 miles upstream when it reaches the opposite shore.

To find out how far upstream the boat is when it reaches the opposite shore, we can break down the motion of the boat into horizontal and vertical components.

First, let's find the velocity of the boat relative to the shore. We can use trigonometry to calculate the horizontal and vertical components of the boat's velocity.

Horizontal component of boat's velocity:
velocity_x = velocity * cos(angle)
velocity_x = 3.30 mi/h * cos(62.5°)

Vertical component of boat's velocity:
velocity_y = velocity * sin(angle)
velocity_y = 3.30 mi/h * sin(62.5°)

Now, let's calculate the time it takes for the boat to cross the river. Since the velocity of the boat relative to the shore is perpendicular to the river's current, the vertical component of the boat's velocity is the same as the current's velocity.

time = distance / velocity_y
time = 0.795 mi / (1.25 mi/h)

Once we have the time, we can find the distance traveled upstream by the boat using the horizontal component of its velocity.

distance_upstream = velocity_x * time

Plug in the calculated values to find the answer.

Velocity component North (across river)

= 3.3 sin 62.5 = 2.93 mi/h
time to cross = .795/2.93 = .272 hr

Velocity component west (upriver)
= 3.3 cos 62.5 - 1.25
= .274 mi /hr

distance upstream = .274*.272 = .0744 mi
*5280 ft.mi = 393 feet