A boy can row 18km. downstream and return a distance of 16 km/hr upstream in 5hrs. If the rate of the current is 6km/hr, What is the boy's rate of rowing in still water?
To find the boy's rate of rowing in still water, we need to use the concept of relative velocity.
Let's assume the boy's rate of rowing in still water is "x" km/hr.
When the boy rows downstream (in the same direction as the current), the effective speed is increased due to the current. The total speed is the sum of the boy's rate of rowing and the speed of the current, which is (x + 6) km/hr.
So, the time taken to row downstream a distance of 18 km can be calculated using the formula:
Time = Distance / Speed
Time = 18 km / (x + 6) km/hr
When the boy rows upstream (against the current), the effective speed is decreased due to the current. The total speed is the difference between the boy's rate of rowing and the speed of the current, which is (x - 6) km/hr.
So, the time taken to row upstream a distance of 16 km can be calculated using the formula:
Time = Distance / Speed
Time = 16 km / (x - 6) km/hr
We are given that the total time for both trips is 5 hours. Therefore, the sum of the time taken downstream and upstream is equal to 5:
18 / (x + 6) + 16 / (x - 6) = 5
Now, we can solve this equation to find the value of "x," which represents the boy's rate of rowing in still water.
To solve this equation, we can multiply every term by the common denominator, which is (x + 6)(x - 6). This will eliminate the denominators:
18(x - 6) + 16(x + 6) = 5(x + 6)(x - 6)
Simplifying the equation:
18x - 108 + 16x + 96 = 5(x^2 - 36)
Combine like terms:
34x - 12 = 5x^2 - 180
Rearrange the equation to form a quadratic equation equal to 0:
5x^2 - 34x - 168 = 0
Now, we can solve this equation using factoring, completing the square, or using the quadratic formula. Once we find the value of "x," we will know the boy's rate of rowing in still water.