A motorboat that travels with a speed of 20 km/hour in still water has traveled 20 km against the current and 180 km with the current, having spent 8 hours on the entire trip. Find the speed of the current of the river.
To find the speed of the current, let's assume the speed of the current is represented by 'c' km/hour.
We know that the motorboat travels against the current at a speed of 20 km/hour, so the effective speed of the boat against the current is (20 - c) km/hour.
Similarly, when the boat is traveling with the current, its effective speed is (20 + c) km/hour.
Now, let's calculate the time it takes to travel against the current and with the current.
Time taken to travel 20 km against the current = Distance / Speed
= 20 / (20 - c) (1)
Similarly, time taken to travel 180 km with the current = Distance / Speed
= 180 / (20 + c) (2)
According to the problem, the total time taken for the entire trip is 8 hours.
So, (1) + (2) = 8
20 / (20 - c) + 180 / (20 + c) = 8
Now, let's solve this equation to find the value of 'c', the speed of the current.
Multiply both sides of the equation by (20 - c)(20 + c) to get rid of the denominators.
20(20 + c) + 180(20 - c) = 8(20 - c)(20 + c)
400 + 20c + 3600 - 180c = 8(400 - c^2)
Simplify the equation:
4000 + 20c - 180c = 3200 - 8c^2
-8c^2 - 20c + 800 = 0
Divide the equation by -4 to simplify further:
2c^2 + 5c - 200 = 0
Factorize the quadratic equation:
(2c - 25)(c + 8) = 0
From here, we have two possible solutions:
1. 2c - 25 = 0 => c = 25/2 = 12.5
2. c + 8 = 0 => c = -8
Since speed cannot be negative in this context, we ignore the negative value of 'c' and conclude that the speed of the current is 12.5 km/hour.
Therefore, the speed of the current of the river is 12.5 km/hour.
speed of the curren --- x km/h
so the speed with the current = 20+x km/h
and speed against the current = 20 - x km/h
time against the current = 20/(20-x)
time with the current = 180/(20+x)
20/(20-x) + 180/(20+x) = 8
solve for x, you will get a quadratic equation