Question

19/A steamer goes downstream from one port to another in 9 hours. It covers the same distance upstream in 10 hours. If the speed of the stream be 1 km/h, find the speed of the steamer in still water and the distance between the ports. ​

Answer 1

Aиѕωєr —

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~ The speed of streamer is 19 km / h , and the distance between the two ports is 180 km .

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Exρℓαиαтiσи –

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Let's strat with what is speed , Speed is a scaler quantity which is defined as the distance travelled by a body or partical per unit time . Generally we deal with two kinds of speed - ínstantaneous and average speed . For uniform motion both are identical .

Here, we would be using average speed in this question. Average speed is defined as the total distance covered divides by the time taken by the body to cover it .

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$\qquad{\pmb{\mathfrak{\longrightarrow \quad Speed_{avg}=\dfrac{distance}{time}}}}$

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Now , starting the solution of above question , let the distance be x km , and speed of streamer be y km/h.

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We are given that :

• while going downstream from one port to another , streamer takes 9 hours.
• It covers the same distance upstream in 10 hours.
• the speed of the stream be 1 km/h

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We know that :

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$\qquad\sf{\dashrightarrow \quad Speed_{avg}=\dfrac{distance}{time}}$

$\qquad\sf{\dashrightarrow \quad time = \dfrac{distance}{speed_{avg}}}$

$\qquad\sf{\dashrightarrow \quad time =\dfrac{distance\: covered\: upstream}{speed_{avg}-speed_{current}}}$

$\qquad\sf{\dashrightarrow \quad 10 =\dfrac{x}{y-1}\qquad\qquad\cdots \cdots ( 1 ) }$

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While going downstream :

$\qquad\sf{\dashrightarrow \quad Speed_{avg}=\dfrac{distance}{time}}$

$\qquad\sf{\dashrightarrow \quad time = \dfrac{distance}{speed_{avg}}}$

$\qquad\sf{\dashrightarrow \quad time =\dfrac{distance\: covered\: upstream}{speed_{avg}+ speed_{current}}}$

$\qquad\sf{\dashrightarrow \quad 9 = \dfrac{x}{y+1}\qquad\qquad\cdots\cdots (2 )}$

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On dividing equation ( 1 ) by ( 2 ) :

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$\qquad\tt{\dashrightarrow \quad \dfrac{10}{9}=\dfrac{\dfrac{x}{y-1}}{\dfrac{x}{y+1}}}$

$\qquad\tt{\dashrightarrow \quad \dfrac{10}{9}=\dfrac{\cancel{x}}{y-1}\times \dfrac{y+1}{\cancel{x}}}$

$\qquad\tt{\dashrightarrow \quad \dfrac{10}{9}=\dfrac{y+1}{y-1}}$

$\qquad\tt{\dashrightarrow \quad 10(y-1)=9(y+1)}$

$\qquad\tt{\dashrightarrow \quad 10y-10=9y+9}$

$\qquad\tt{\dashrightarrow \quad 10y-9y=9+10}$

$\qquad{\pmb{\tt{\dashrightarrow \quad y = 19 \:km/hr}}}$

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Substituting , the value of y in equation ( 1 ) :

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$\qquad\tt{\dashrightarrow \quad 10 = \dfrac{x}{y-1}}$

$\qquad\tt{\dashrightarrow \quad 10 = \dfrac{x}{19-1}}$

$\qquad\tt{\dashrightarrow \quad 10=\dfrac{x}{18}}$

$\qquad\tt{\dashrightarrow \quad x = 10\times 18}$

$\qquad{\pmb{\tt{\dashrightarrow \quad x = 180\:km }}}$

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❝ Hence, the speed of the streamer is 19 km/h and the distance covered is 180 km.❞

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