Question

Example 15: A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that the speed of the stream be x km per hour.

Given that, Speed of motor boat in stil water = 18 km per hour

So, speed of motor boat in upstream = 18 - x km per hour

Speed of the motor boat in downstream = 18 + x km per hour.

Now,

Distance covered in downstream = 24 km

Time taken in downstream = $\dfrac{24}{18+x}$

Distance covered in upstream = 24 km

Time taken in downstream = $\dfrac{24}{18-x}$

According to statement, A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot

So,

$\rm \: \dfrac{24}{18 - x} - \dfrac{24}{18 + x} = 1$

$\rm \: \dfrac{24(18 + x) - 24(18 - x)}{(18 - x)(18 + x)} = 1$

$\rm \: \dfrac{24 \times 18 +24 x - 24 \times 18 + 24x}{(18 - x)(18 + x)} = 1$

$\rm \: \dfrac{48x}{(18 - x)(18 + x)} = 1$

$\rm \: \dfrac{48x}{324 - {x}^{2} } = 1$

$\rm \: 48x = 324 - {x}^{2}$

$\rm \: {x}^{2} + 48x - 324 = 0$

$\rm \: {x}^{2} + 54x - 6x - 324 = 0$

$\rm \: x(x+ 54) - 6(x + 54) = 0$

$\rm \: (x+ 54)(x - 6) = 0$

$\rm\implies \:x = 6 \: \: or \: \: x \: = \: - \: 54 \: \: \{rejected \}$

So,

$\rm\implies \:\boxed{ \rm{ \:Speed\:of\:stream = 6 \: km \: per \: hr \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac

Answer 2

Step-by-step explanation:

$\mathbb\red{ \tiny A \scriptsize \: N \small \:S \large \: W \Large \:E \huge \: R}$

Let the speed of the stream be x km / h. Therefore, the speed of the boat upstream =(18 - x) km / h and the speed of the boat downstream =(18 + x) km / h .

The time taken to go upstream = distance/speed

$\rm =\dfrac{24}{18+x}$

Similarly, the time taken to go downstream

$\rm=\dfrac{24}{18-x}$

According to the question,

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

$\begin{array}{l} \rm \Rightarrow 24(18+x)-24(18-x)=(18-x)(18+x) \\ \\ \rm \Rightarrow x^{2}+48 x-324=0 \\ \\ \boxed{ \red{\rm \Rightarrow x = 6}} \rm \: \: \: or \: \: \: \: - 5 4 \end{array}$

Since x is the speed of the stream, it cannot be negative. So, we ignore the root x =-54. Therefore, x =6 gives the speed of the stream as 6 km / h.