A fisherman traveled in a boat from point N upstream. After having traveled 6 km, he stopped rowing and 2 hour 40 min after first leaving N, he was brought back to N by the current. Knowing that the speed of the boat in still water is 9 km/hour, find the speed of the river’s current.
speed of current ---- x km/h
so speed against the current = 9-x km/h
speed drifting ----- x km/h
time going against the current = 6/(9-x) hrs
time drifting back down = 6/x hrs
total time:
6/(9-x) + 6/x= 2 2/3 = 8/3
multiply each term by 3x(9-x) and solve for x
To find the speed of the river's current, we can use the concept of relative speed.
Let's assume the speed of the river's current is "x" km/hour. Since the fisherman is traveling upstream, the effective speed of the boat will be the speed of the boat in still water (9 km/hour) minus the speed of the current (x km/hour).
Now, let's analyze the trip:
The fisherman traveled 6 km upstream at the effective speed of the boat, which is (9 - x) km/hour.
The time taken to travel this distance can be calculated by dividing the distance (6 km) by the effective speed ((9 - x) km/hour):
Time taken = Distance / Speed
6 / (9 - x)
After this, the fisherman stops rowing. Since he is no longer rowing, his effective speed is now equal to the speed of the river's current, which is "x" km/hour.
Next, we are given that it took him 2 hours and 40 minutes (or 2 + 40/60 = 2.67 hours) to be brought back to point N by the current.
To calculate the time taken to be brought back to N by the current, we use the same formula as before:
Time taken = Distance / Speed
6 / x
We are given that the total time for the trip is 2 hours and 40 minutes (or 2.67 hours).
So, the total travel time can be calculated by adding the time taken upstream and the time taken downstream:
total time = 6 / (9 - x) + 6 / x
2.67 = 6 / (9 - x) + 6 / x
Now, we have an equation we can solve to find the value of "x" (the speed of the river's current).
To solve this equation, we can multiply both sides by (9 - x)(x) to eliminate the denominators:
2.67 * (9 - x)(x) = (6)(x) + (6)(9 - x)
Expanding and simplifying:
2.67x(9 - x) = 6x + 54 - 6x
Simplifying further:
2.67x(9 - x) = 54
Now, we can divide both sides by 2.67 to solve for "x":
(9 - x) * x = 54 / 2.67
x^2 - 9x + 54/2.67 = 0
This is a quadratic equation, which we can solve using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values:
x = (-(-9) ± √((-9)^2 - 4(1)(54/2.67))) / 2(1)
Simplifying:
x = (9 ± √(81 - 229.91)) / 2
x = (9 ± √(-148.91)) / 2
Since we cannot take the square root of a negative number under the real number system, there are no real solutions to this equation.
Therefore, there is no valid speed for the river's current that satisfies the given conditions.