Answer:
The speed of the river's current is 4.5 km/hr.
Step-by-step explanation:
Given : A fisherman traveled in a boat from point N upstream.
After having traveled 6 km, he stopped rowing and 2 hour 40 min after first leaving N, he was brought back to N by the current.
Knowing that the speed of the boat in still water is 9 km/hour
To find : The speed of the river’s current.
Solution :
Let the speed of current = x km/h
Speed of boat = 9 km/h
Thus, the speed in upstream = (9 - x) km/h
Also, the speed in downstream = x km/h (because the boat is stationary in downstream)
Total distance = 6 km ( from N to the point he reached )
Total time = 2 hours 40 minutes
\(\text{2 hours 40 minutes}=2\frac{40}{60}=2 \frac{2}{3} \\\\\text{2 hours 40 minutes}=\frac{8}{3} \text{ hours}\)
We know, \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)
According to the question,
\(\frac{6}{9-x}+\frac{6}{x}=\frac{8}{3}\)
Taking LCM and 6 common
\(6[\frac{x+9-x}{(9-x)x}]=\frac{8}{3}\)
Take 6 to another side,
\(\frac{9}{9x-x^2}=\frac{8}{3\times 6}\)
Cross multiply,
\(9\times 18=8\times(9x-x^2)\)
\(162=72x-8x^2\)
Taking 4 common and cancel out
\(4x^2-36x+81=0\)
Solve quadratic equation by middle term split,
\(4x^2-18x-18x+81=0\)
\(2x(2x-9)-9(2x-9)=0\)
\((2x-9)(2x-9)=0\)
\(\Rightarrow 2x-9=0\)
\(\Rightarrow x=\frac{9}{2}=4.5\)
Therefore, The speed of the river's current is 4.5 km/hr.
