A motor boat whose speed is $18\ km/hr$ in still water $1\ hr$ more to go $24\ km$ upstream then to return downstream to the same spot. Find the speed of the stream.

Given: A motor boat whose speed is $18\ km/hr$ in still water $1\ hr$ more to go $24\ km$ upstream then to return downstream to the same spot.

To do: To find the speed of the stream.

Solution:

Let the speed of stream be $x\ km/hr$
 
Now, for upstream: 

speed $= (18-x)\ km/ hr$

 time taken $=(\frac{24}{18-x})\ hr$

Now, for downstream: 

speed $=(18+x)\ km/hr$

time taken $=(\frac{24}{18+x})\ hr$

Given that,

$(\frac{24}{18-x})=(\frac{24}{18+x})+1$

$\Rightarrow (\frac{24}{18-x})-(\frac{24}{18+x})=1$

$\Rightarrow 24(\frac{1}{18-x}-\frac{1}{18+x})=1$

$\Rightarrow 24(\frac{18+x-18+x}{(18-x)(18+x)})=1$

$\Rightarrow 24(\frac{2x}{324-x^{2}})=1$

$\Rightarrow 48x=324-x^{2}$

$\Rightarrow x^{2}+48x-324=0$

$\Rightarrow x^{2}+54x-6x-324=0$

$\Rightarrow x( x+54)-6( x+54)=0$

$\Rightarrow ( x+54)( x-6)=0$

If $x+54=0$

$\Rightarrow x=-54$

If $x-6=0$

$\Rightarrow x=6$

$\Rightarrow$ Speed can't be negative. we reject $x=-54$

Therefore, speed of the stream $= 6\ km/hr$
 

Updated on: 10-Oct-2022

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