Question

a motorboat whose speed is 18 kilometre per hour in still water takes 1 hour more to go 24 km upstream then a return downstream to the same spot find the speed of the stream​

Answer 1

${\pmb{\underline{\sf{ Required \ Solution ... }}}}$

• Motor Boat's speed = 18 km/hr
• Distance = 24 km
• Boat takes 1 hr more to go upstream than downstream.

Let the Speed of the stream be x

• For Upstream = (18 - x) km/hr
• For Downstream = (18 + x) km/hr

As we know that:

$\colon\implies{\sf{ t_1{(Upstream)} = \dfrac{24}{18-x} \ hr}} \\ \\ \colon\implies{\sf{ t_2{(Downstream)} = \dfrac{24}{18+x} \ hr }}$

So, We have to subtract coz we've as:

$\circ \ {\underline{\pmb{\sf{ According \ to \ Question: }}}} \\ \\ \colon\implies{\sf{ \dfrac{24}{18-x} - \dfrac{24}{18+x} = 1 }} \\ \\ \colon\implies{\sf{ \dfrac{24(18+x)-24(18-x)}{(18-x)(18+x)} = 1 }} \\ \\ \colon\implies{\sf{ \dfrac{432+24x-432+24x}{(18)^2-x^2} = 1}} \\ \\ \colon\implies{\sf{ \dfrac{48x}{324-x^2} = 1 }} \\ \\ \colon\implies{\sf{ 48x = 324 - x^2 }} \\ \\ \colon\implies{\sf{ x^2+48x - 324 = 0 }}$

Now, We have to apply Quadratic Equation to get the desired value of the Variable as:-

$\circ {\pmb{\underline{\boxed{\sf\gray{ x = \dfrac{ -b \pm \sqrt{b^2 - 4ac} }{2a } }}}}} \\ \\ \colon\implies{\sf{ x = \dfrac{ -48 \pm \sqrt{(-48)^2 - 4 \times 1 \times (-324)} }{2 \times 1 } }} \\ \\ \colon\implies{\sf{ x = \dfrac{ -48 \pm \sqrt{2304+1296} }{2} }} \\ \\ \colon\implies{\sf{ x = \dfrac{ -48 \pm \sqrt{3600} }{2} }} \\ \\ \colon\implies{\sf{ x = \dfrac{ -48 \pm 60 }{2} }} \\ \\ \colon\implies{\sf{ x = \dfrac{ - 48 + 60}{2} \ and \ \dfrac{-48-60}{2} }} \\ \\ \colon\implies{\sf{ x = \cancel{ \dfrac{12}{2} } \ and \ \cancel{ \dfrac{-108}{2} } }} \\ \\ \colon\implies{\underline{\boxed{\sf{ x = 6 \ and \ -54 }}}}$

The Speed can't be Negative (—ve).

Hence,

${\pmb{\underline{\sf{ The \ Speed \ of \ the \ Stream \ is \ 6 \ km/hr. }}}}$ $\bigstar$

Answer 2

Let the speed of the stream be x km\hr.

The speed of the boat upstream = (18 - x) km/hr

The speed of the boat downstream = (18 + x) km/hr

Distance = 24 km

As given in the question,

Time for upstream = 1 + Time for downstream

24/(18 - x) = 1 + 24/(18 + x)

24/(18 - x) - 24/(18 + x) = 1

x2 + 48x - 324 = 0

(x + 54)(x - 6) = 0

x ≠ - 54 as speed cannot be negative.

x = 6

The speed of the stream = 6 km/hr