Question

A motor boat heads upstream a distance of 24km on a river whose current is running at 3km per hour.
The trip up and back takes 6 hours. Assuming that the motor boat maintained a constant speed, what was
its speed?(Conn)
​

Answer 1

AnswEr :

9km/hrs.

$\bf{\red{\underline{\underline{\bf{Given\::}}}}}$

A motor boat heads upstream a distance of 24 km on a river whose current is running at 3 km per hours. The trip up and back takes 6 hours.

$\bf{\red{\underline{\underline{\bf{To\:find\::}}}}}$

The speed of the stream.

$\bf{\red{\underline{\underline{\bf{Explanation\::}}}}}$

Let the speed of the stream be r

Let $\sf{t_{1}}$ and $\sf{t_{2}}$ be the time for the upstream and downstream.

$\bf{\blue{\underline{\underline{\tt{1_{st}\:Case\::}}}}}$

$\bf{We\:have}\begin{cases}\sf{A\:motor\:boat\:speed\:for\:Upstream=(r-3)km/hrs}\\ \sf{Distance\:(d)=24km}\\ \sf{Time\:(t)=t_{1}}\end{cases}}$

Formula use :

$\bf{\boxed{\sf{Time=\frac{Distance}{Speed}}}}$

$\leadsto\sf{t_{1}=\dfrac{24}{r-3} }}$

$\bf{\blue{\underline{\underline{\tt{2_{nd}\:Case\::}}}}}$

$\bf{We\:have}\begin{cases}\sf{A\:motor\:boat\:speed\:for\:downstream=(r+3)km/hrs}\\ \sf{Distance\:(d)=24km}\\ \sf{Time\:(t)=t_{2}}\end{cases}}$

$\leadsto\sf{t_{2}=\dfrac{24}{r+3} }$

$\bigstar\:\bf{\underline{\underline{\bf{According\:to\:the\:question\::}}}}}$

$\mapsto\sf{\dfrac{24}{r-3} +\dfrac{24}{r+3} =6}\\\\\\\mapsto\sf{\dfrac{24(r+3)+24(r-3)}{(r-3)(r+3)} =6}\\\\\\\mapsto\sf{\dfrac{24r\cancel{+72}+24r\cancel{-72}}{(r-3)(r+3)} =6}\\\\\\\mapsto\sf{\dfrac{48r}{(r-3)(r+3)} =6}\\\\\\\mapsto\sf{48r=6(r-3)(r+3)}\\\\\\\mapsto\sf{48r=6(r^{2}\cancel{+3r} \cancel{-3r}-9)}\\\\\\\mapsto\sf{48r=6(r^{2} -9)}\\\\\\\mapsto\sf{48r=6r^{2} -54}\\\\\\\mapsto\sf{6r^{2} -48r-54=0}\\\\\\\mapsto\sf{6(r^{2} -8r-9)=0}$

$\mapsto\sf{r^{2} -8r-9=\cancel{\dfrac{0}{6} }}\\\\\\\mapsto\sf{r^{2} -8r-9=0}\\\\\\\mapsto\sf{r^{2} -9r+r-9=0}\\\\\\\mapsto\sf{r(r-9)+1(r-9)=0}\\\\\\\mapsto\sf{(r-9)(r+1)=0}\\\\\\\mapsto\sf{r-9=0\:\:\:\:Or\:\:\:\:r+1=0}\\\\\\\mapsto\sf{\red{r=9\:\:\:\:Or\:\:\:\:r=-1}}$

We know that negative value isn't acceptable.

Thus;

$\star\underbrace{\sf{The\:speed\:of\:the\:stream\:is=\:r=\:9\:km/hrs}}}}$

Answer 2

$</p><p>\tt{\pink{\huge{Hello!!! }}}$

$</p><p>\tt{\red{Given \: - }}$

• Distance = 24 km.

• Speed of Current = 3 kmph.

• Total time = 6 hrs.

______________

Assume the speed of the motorboat to be X kmph.

$</p><p>\tt{\orange{Speed_{Upstream} = X - 3 }}$

$</p><p>\tt{\purple{Speed_{Downstream} = X + 3 }}$

o

Hence :

$\tt{ \red{ \implies{\: \dfrac{24}{x - 3} + \dfrac{24}{x + 3} \: = 6 \: }}}$

Solving We Get :

$</p><p>\tt{\purple{X = 9 km/hr }}$