Question

The speed of a boat in still water is \[11{\text{ }}km/hr\]. It can go 12 km upstream and return downstream to its original point in \[2{\text{ }}hr{\text{ }}45{\text{ }}min\]. Find out the speed of the stream.

Answer 1

Hint: The boat will move with the different speeds in upstream and downstream. In upstream it will slow down due to the speed of the river opposing the actual motion of the boat and downstream the same motion of the boat will get supported by the movement of water. This will lead to different time intervals to travel upstream and downstream.

Complete step by step answer:
Consider the velocity of water is $x$. The velocity of the boat is $ 11\,km/hr$. The time to reach the end of upstream $T$. The time to reach the end downstream is $t$. The distance upstream and downstream is $12\,km$. From the basic equation we know that $ speed = \dfrac{d}{{time}}$
Now for upstream,
Let the velocity be
$11 - x \\
\Rightarrow T = \dfrac{{12}}{{11 - x}}...(i) \\ $
For downstream let the velocity be
$11 + x \\
\Rightarrow t = \dfrac{{12}}{{11 + x}}...(ii) \\ $
As we know from the question total time is \[2hr{\text{ }}45min\]
$ 2 + \dfrac{3}{4} \\
\Rightarrow 2.75hr \\ $
Now adding (i) and (ii) we get
$2.75 = \dfrac{{12}}{{11 - x}} + \dfrac{{12}}{{11 + x}} \\
\Rightarrow \dfrac{{2.75}}{{12}} = \dfrac{{22}}{{11 - {x^2}}} \\ $
Cross multiplying and solving the quadratic equation
$ 11 - {x^2} = \dfrac{{264}}{{2.75}} \\
\Rightarrow {x^2} = 96 - 11 \\
\Rightarrow {x^2} = 85$
Taking the square root we get
$\therefore x = 9.21\,km/hr$

The speed of the stream is \[9.21\,km/hr\].

Note:As the speed upstream is decreased and the boat will move slowly, the time required will be more. As the speed is increased downstream the distance will be covered faster i.e. less time will be required. In still water the stream speed is zero.