Question

A motor boat has a speed of 18 $\frac{km}{hr}$ in still water. It takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream in $km/hr.$
[3 Marks]

Answer 1

Let the speed of the stream be '$x$' and the time required to travel upstream be '$t$'.

$Time=\frac{Distance}{Speed}$
[.5 mark]

Speed of boat in upstream
= Speed of boat - Speed of stream
= 18 - $x$

and, speed of boat in downstream
= speed of boat + speed of stream
= 18 + $x$ [.5 Mark]

$\therefore$ From the given data,

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

$t+1=\frac{24}{(18-x)}$------------(2)

Subtracting (1) from (2), we get

$1=24[\frac{1}{18-x}-\frac{1}{18+x}]$

$\Rightarrow 1=24\left[\frac{2x}{({18}^2-x^2)}\right]$

$\Rightarrow {18}^2-x^2=48x$

$\Rightarrow x^2+48x-{18}^2=0$ [1 Mark]

Now, the equation $x^2+48x-{18}^2=0$ can also be written as:

$x^2+2\left(24\right)x+{24}^2-{24}^2-{18}^2=0$

$\Rightarrow$$(x+24{)}^2-900=0$

$\Rightarrow$$(x+24{)}^2-{30}^2$= 0

$\Rightarrow$$(x+24{)}^2={30}^2$

$\Rightarrow$$x+24=\pm 30$ (Taking square root on both sides)

$\Rightarrow$ $x=-54$ or $x=6$

The speed of the stream cannot be negative.

So, the speed of the stream $=6km/hr$
[1 Mark]