To solve this problem, we need to determine the distance from the starting point on the first bank to the point of reaching on the second bank, taking into account the river's flow.
Here's a step-by-step solution:
1. Identify the speeds involved:
- Speed of the man swimming in still water: [tex]\( 6 \, \text{km/h} \)[/tex]
- Speed of the river flow: [tex]\( 3 \, \text{km/h} \)[/tex]
2. Determine the resultant velocity of the man swimming across the river:
To swim directly across the river, the man needs to compensate for the river's flow. We can use the Pythagorean theorem to find the resultant velocity ([tex]\(V_r\)[/tex]):
[tex]\[
V_r = \sqrt{V_{\text{swim}}^2 - V_{\text{river}}^2}
\][/tex]
Substituting the given values:
[tex]\[
V_r = \sqrt{6^2 - 3^2} = \sqrt{36 - 9} = \sqrt{27} \approx 5.196 \, \text{km/h}
\][/tex]
3. Calculate the time taken to cross the river:
- Width of the river: [tex]\(1 \, \text{km}\)[/tex]
- Time taken ([tex]\(t\)[/tex]) to cross the river is given by the width of the river divided by the resultant velocity:
[tex]\[
t = \frac{1 \, \text{km}}{5.196 \, \text{km/h}} \approx 0.192 \, \text{hours}
\][/tex]
4. Determine the distance traveled along the river due to the current's flow:
This distance is determined by the speed of the river and the time taken to cross the river:
[tex]\[
\text{Distance along the river} = V_{\text{river}} \times t = 3 \, \text{km/h} \times 0.192 \, \text{hours} \approx 0.577 \, \text{km}
\][/tex]
Thus, the distance from the starting point on the first bank to the point of reaching on the second bank is approximately [tex]\( 0.577 \, \text{km} \)[/tex].
Among the given options, the closest match is:
[tex]\[
\frac{\sqrt{5}}{2} \, \text{km}
\][/tex]
We can approximate:
[tex]\[
\sqrt{5} \approx 2.236
\][/tex]
So,
[tex]\[
\frac{\sqrt{5}}{2} \approx \frac{2.236}{2} \approx 1.118 \, \text{km}
\][/tex]
It seems there's a discrepancy in the provided options. Strictly based on the given options, none exactly match [tex]\(0.577 \, \text{km}\)[/tex]. But for practical purposes, [tex]\(\frac{\sqrt{5}}{2}\)[/tex] is the closest provided choice, though it simplifies to approximately 1.118 km. If the options might have a mistake, the closest answer seems to be missing from the provided options.
