a boat crosses a 2 km wide river by starting out at the south shore and heading directly for the north shore if the river flows at 0.8 m/s and the boat can go at 1.4 m/s in still water, where on the opposite shore will the boat land?
What direction is the river flowing?
I found the answer, it is 1142.86 meters.
I assume that the river was flowing East.
so, the angle = arctan( .8 / 1.4 )
= 29.74 degrees.
then tan (29.74) = offset / 2000 m
= 1142.86 m
To find where the boat will land on the opposite shore, we need to calculate the total distance the boat will be drifted downstream by the river's flow during the crossing.
Let's use the following variables:
- D: Total distance the boat needs to cross the river (2 km or 2000 m).
- R: Velocity of the river's flow (0.8 m/s).
- B: Velocity of the boat in still water (1.4 m/s).
- T: Time taken by the boat to cross the river.
Based on the given information, we can set up the following equation:
Distance traveled by the boat in still water = Distance across the river + Drift downstream
Distance across the river = 2 km or 2000 m.
Using the formula distance = velocity × time, we can now write the equation:
B * T = D + R * T
Since the boat is heading directly from the south shore to the north shore, the distance traveled in still water is equal to the distance across the river, so we can write the equation as:
B * T = 2000 m + 0.8 m/s * T
Now, let's solve the equation to find the value of T:
1.4 m/s * T = 2000 m + 0.8 m/s * T
1.4 m/s * T - 0.8 m/s * T = 2000 m
0.6 m/s * T = 2000 m
T = 2000 m / 0.6 m/s
T ≈ 3333.33 seconds
Now that we know the time it takes for the boat to cross the river (approximately 3333.33 seconds), we can find the distance the boat will be drifted downstream by the river's flow:
Drift downstream = R * T
Drift downstream = 0.8 m/s * 3333.33 seconds
Drift downstream ≈ 2666.67 meters
Therefore, the boat will land approximately 2666.67 meters downstream from the starting point on the opposite shore.