Question

The boat sails down the river for 10 km and then up the river for 6 km. The speed of the river flow is 1km/h. What is the minimum speed of the boat for the trip to take a maximum of 4 hrs.?

Answer 1

Hint: When we go in the direction of the river then our speed increases by speed of river. Similarly we go opposite of the river then our speed decreases by speed of the river. Now you know distance, speed. So, calculate time, up the river and down the river. Now add them and equate the sum to the given value.

Complete step-by-step answer:
Given speed in the river in the question assumed to be S, is:
\[S = 1\,km/hr\]
Let us assume the speed of the boat to satisfy the given condition is x.
Speed of boat \[ = x\,km/hr\]
Down the river means the boat moves in the direction of the river.
Up the river means the boat moves in the opposite direction of the river.
So, while down the river, the water supports the boat:
Speed down \[ = \left( {x + 1} \right)\,km/hr\]
While going up the river, the water flow opposes the boat:
Speed up \[ = \left( {x - 1} \right)\,km/hr\]
As we know, \[time = \dfrac{{dis\tan ce}}{{speed}}\]
We know boat sails 10 km down the river, we get time as:
Time down \[ = \dfrac{{10}}{{\left( {x + 1} \right)}}\,hrs\]
We know boat sails 6 km up the river, we get time as:
Time up \[ = \dfrac{6}{{\left( {x - 1} \right)}}\,hrs\]
It is given in question maximum time taken is 4 hrs.:
Total Time= Time down + Time up
By substituting all the values, we get the equation as:
\[\dfrac{{10}}{{x + 1}}\, + \dfrac{6}{{x - 1}} = 4\,\]
By taking least common multiple and cross multiplication, we get:
\[10\left( {x - 1} \right) + 6\left( {x + 1} \right) = 4\left( {x + 1} \right)\left( {x - 1} \right)\]
By expanding all the terms on both sides, we get it as:
\[10x - 10 + 6x + 6 = 4{x^2} - 4\]
By simplifying whole equation, we get it as:
\[4{x^2} - 16x = 0\]
By taking 4x as common, we can write it in form of:
\[4x\left( {x - 4} \right) = 0\]
By equating both equations to zero, we get it as:
\[x = 0\,\,,\,\,x - 4 = 0\]
As x is speed 0 is not meaningful. So we take another \[x = 4\].
Therefore the speed of the boat must be 4 km/hr to satisfy condition.
Note: Without solving quadratic you can compare terms in the equations on both sides to get the same result. The idea of the effect of speed of the river is very important, be careful in those steps. Do not confuse between up and down the river, down means along and up means opposite. Remember them.