Question

A motorboat whose speed is 9 km per hour in still water goes 15 km downstream and come back in a total time of 3 hour and 45 minutes find the speed of the stream

Answer 1

$<b>$
Let the speed of the stream be x km/hr.

Speed downstream = (9 + x) km/hr,
Speed upstream = (9 - x) km/hr

Given,
Boat took 3hrs 45 min
viz., $3+ \frac{45}{60} = 3+ \frac{3}{4} = \frac{15}{4}$ hrs to travel back to same point.

We have,
$(time\: taken)_{downstream} + (time\: taken)_{upstream} = (time\: taken)_{total}$

$time = \frac{distance}{velocity}$

So,
$\frac{15}{9+x} - \frac{15}{9-x} = \frac{15}{4}$

$\implies \frac{15×18}{81-x^{2}} =\frac{9}{2}$

$\implies 81-x^{2} = 18×4$

$\implies 81 - x^{2} = 72$

$\implies x^{2} = 81-72$

$\implies x^{2} = 9$

$\implies x = 3$

Therefore ,
Speed of the stream = 3 kmph

Answer 2

Let the speed of the stream be x km/h

$\therefore$ Speed Downstream = (15+x) km/h

Speed upstream = (15-x)km/h

Time taken to travel 30 km downstream = $\sf{\frac{30}{15+x}h}$

Time taken to travel 30 km upstream = $\sf{\frac{30}{15-x}h}$

Given, total time taken =

$\sf\green{4 \: hours \: 30 \: mins. = (4 + \frac{1}{2} )h = \frac{9}{2} h}$

$\longrightarrow$$\sf\red{ \frac{30}{15 + x} + \frac{30}{15 - x} = \frac{9}{2}}$

$\longrightarrow$$\sf\red{\frac{10}{15 + x} + \frac{10}{15 - x} = \frac{3}{2} }$

$\longrightarrow$$\sf\red{\frac{10(15 - x + 15 + x)}{(15 + x)(15 - x)} = \frac{3}{2} }$

$\longrightarrow$$\sf\red{10 \times 30 \times 2 = 3(225 - {x}^{2} )}$

$\longrightarrow$$\sf\red{200 = 225 \times {x}^{2}}$

$\longrightarrow$$\sf\red{{x}^{2} = 25 \: or \: x = ±5}$

As speed cannot be negative, x ≠-5, so x = 5

Hence, the speed of the stream = 5km/h