Question

The current of a stream runs at the rate of 4 km an hour. A boat goes 6 km and back to the starting point in 2 hours. The speed of the boat in still water is:

A. 6km/hr

B. 7.5km/hr

C. 8 km/hr

D. 6.8 km/hr

Answer 1

Hint: In this problem, first we need to form the quadratic equation using the time and speed formula. Next, solve the quadratic equation using the middle term factorization method.

Complete step-by-step answer:

The speed of current of a stream is 4km/hr.

Consider, the speed of the boat in still water be \[x\] km/hr. 

We know that Speed is given by the formulae Distance traveled by time taken.

i.e S = D/T. Therefore Time can be written as 

Time  = Distance / Speed 

Since, the boat goes 6 km along the stream and back to the starting point in 2 hours, therefore,

$\dfrac{6}{{x + 4}} + \dfrac{6}{{x - 4}} = 2 $

$   \Rightarrow \dfrac{{6\left( {x - 4} \right) + 6\left( {x + 4} \right)}}{{\left( {x + 4} \right)\left( {x - 4} \right)}} = 2 $

$   \Rightarrow \dfrac{{6x - 24 + 6x + 24}}{{{x^2} - 16}} = 2 $

$   \Rightarrow 12x = 2{x^2} - 32 $

$   \Rightarrow 2{x^2} - 12x - 32 = 0 $

Further, solve the above quadratic equation.

$ 2{x^2} - 12x - 32 = 0 $

$   \Rightarrow {x^2} - 6x - 16 = 0 $

$   \Rightarrow {x^2} - \left( {8 - 2} \right)x - 16 = 0 $

$   \Rightarrow {x^2} - 8x + 2x - 16 = 0 $

$   \Rightarrow x\left( {x - 8} \right) + 2\left( {x - 8} \right) = 0 $

$   \Rightarrow \left( {x + 2} \right)\left( {x - 8} \right) = 0 $

$   \Rightarrow x =  - 2\,\,{\text{or}}\,\,8 $

Since, the speed cannot be negative; the speed of the boat in still water is 8km/hr.

Thus, the option (C) is the correct answer.

Note: The speed of the boat is high in the direction of the stream, whereas the speed of the boat is less against the direction of stream. The ratio of the distance to speed represents time.