Question

A motorboat whose speed in still water is \[18km/hr\], 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

Hint: Here, we have to find the speed of the stream. The moving water in a river is called a stream. Under still water circumstances the water is considered to be stationary and the speed of the water is zero. We will find the upstream and downstream speed of the motorboat using the formula. Then we will use the given condition, apply the time and distance formula to form an equation. We will then solve the equation to convert it into a quadratic equation. We will further solve the equation to get the speed of the stream.
Formula used:
 We will use the following formulas:
1.Upstream \[ = \]Speed of the boat in still water \[ - \] Speed of the stream
2.Downstream \[ = \]Speed of the boat in still water \[ + \] Speed of the stream
3.\[{\rm{Time}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\]

Complete step-by-step answer:
Let the speed of the stream be \[x\] km/hr.
Speed of the motorboat in still water \[ = 18km/hr\]
By using the formula Upstream \[ = \]Speed of the boat in still water \[ - \] Speed of the stream, we get
Speed of the motorboat Upstream \[ = 18 - x\]km/hr
By using the formula, Downstream \[ = \]Speed of the boat in still water \[ + \] Speed of the stream, we get
Speed of the motorboat Downstream \[ = 18 + x\] km/hr
Now we know that the distance is 24 km.
It is given that Time taken by Upstream \[ = \] Time taken by downstream \[ + 1\]
By using the formula \[{\rm{Time}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\], we have
So, \[\dfrac{{{\rm{Distanc}}{{\rm{e}}_{{\rm{upstream}}}}}}{{{\rm{Spee}}{{\rm{d}}_{{\rm{upstream}}}}}}{\rm{ = }}\dfrac{{{\rm{Distanc}}{{\rm{e}}_{{\rm{downstream}}}}}}{{{\rm{Spee}}{{\rm{d}}_{{\rm{downstream}}}}}}{\rm{ + 1}}\]
Substituting all the above values, we get
\[ \Rightarrow \dfrac{{24}}{{18 - x}} = \dfrac{{24}}{{18 + x}} + 1\]
By taking LCM on R.H.S, we ger
\[ \Rightarrow \dfrac{{24}}{{18 - x}} = \dfrac{{24 + 18 + x}}{{18 + x}}\]
By cross multiplying and simplifying the equation, we get
\[ \Rightarrow 24(18 + x) = (42 + x)(18 - x)\]
Multiplying the terms, we get
\[ \Rightarrow 432 + 24x = 756 - 42x + 18x - {x^2}\]
Rewriting the equation, we have
\[ \Rightarrow 432 + 24x - 756 + 42x - 18x + {x^2} = 0\]
Adding and subtracting the like terms, we get
\[ \Rightarrow {x^2} + 48x - 324 = 0\]
By factoring the equation, we get
\[ \Rightarrow (x - 6)(x + 54) = 0\]
 \[ \Rightarrow x = 6\] or
\[ \Rightarrow x = - 54\]
Since, the speed of the stream cannot be negative, \[x = - 54\] is neglected.
Therefore, the speed of the stream is \[6km/hr\].

Note: We need to remember the formulas that are used to calculate the speed of the still water, speed of the upstream, and speed of the downstream. If the boat is flowing in the opposite direction to the stream then we call it upstream. The net speed of the boat while going opposite to the direction of the stream is called the upstream speed. If the boat is flowing along the direction of the stream then it is said to be downstream. The net speed of the boat while going along the direction of the stream is called downstream speed.