To solve this problem, we need to determine the angle [tex]\(\theta\)[/tex] that the man should swim relative to the direction of the stream to cross the river along the shortest path.
Given:
- The speed of the man in still water, [tex]\(v_m\)[/tex] is 15 km/h.
- The speed of the current, [tex]\(v_c\)[/tex] is 10 km/h.
The shortest path across the river is attained when the man's effective velocity directly crosses the river perpendicular to the direction of the stream. To achieve this, the man has to swim at an angle [tex]\(\theta\)[/tex] relative to the direction of the flow of the current.
One approach involves using trigonometry, specifically the sine function. The relationship between the speeds and the angle [tex]\(\theta\)[/tex] can be given as:
[tex]\[ \sin(\theta) = \frac{v_c}{v_m} \][/tex]
Let's plug in the given values:
[tex]\[ \sin(\theta) = \frac{10 \text{ km/h}}{15 \text{ km/h}} \][/tex]
[tex]\[ \sin(\theta) = \frac{2}{3} \][/tex]
To find the angle [tex]\(\theta\)[/tex], we take the inverse sine (also known as arcsine) of [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ \theta = \sin^{-1}\left(\frac{2}{3}\right) \][/tex]
Now, to express this angle in degrees:
[tex]\[ \theta \approx 41.81^\circ \][/tex]
Hence, the correct answer is that the man should swim at an angle [tex]\(\theta = \sin^{-1}\left(\frac{2}{3}\right)\)[/tex] with the direction of the stream.
Therefore, the correct option is:
1) [tex]\(\sin^{-1}\left(\frac{2}{3}\right)\)[/tex]
