A canoeist can paddle 12 km upstream and 12 km back downstream in the same amount of time as she can paddle 25 km in still water. If the rate of the current is 2 km/hour, what is her rate in still water?
Let her rate in still water be x km/h
time in still water = 25/x hours
time to go upstream = 12/(x-2)
time to go downstream = 12/(x+2)
12/(x-2) + 12/(x+2) = 25/x
(12x + 24 + 12x - 24)/(x^2 - 4) = 25/x
24x/(x^2 - 4) = 25/x
25x^2 - 100 = 24x^2
x^2 = 100
x = ± √10
rate in still water is 10 km/h
To solve this problem, we can use the concept of relative speed.
Let's assume the canoeist's rate in still water is x km/hour.
When the canoeist is paddling upstream, against the current, her effective speed is reduced by the speed of the current. So, her effective speed is (x - 2) km/hour.
When the canoeist is paddling downstream, with the current, her effective speed is increased by the speed of the current. So, her effective speed is (x + 2) km/hour.
Now, let's calculate the time taken for each leg of the journey:
Time taken to paddle upstream = Distance / Speed = 12 / (x - 2) hours
Time taken to paddle downstream = Distance / Speed = 12 / (x + 2) hours
According to the problem, these two times are equal:
12 / (x - 2) = 12 / (x + 2)
Now, let's solve this equation to find the value of x:
Cross multiplying, we get:
12(x + 2) = 12(x - 2)
Expanding the equation:
12x + 24 = 12x - 24
Simplifying the equation:
24 = -24
This is not possible. The equation has no solution.
Hence, there is no rate in still water that satisfies the given conditions.
Therefore, there is no possible answer to this problem.
To solve this problem, let's assume that the canoeist's rate in still water is x km/hour.
When the canoeist is paddling upstream (against the current), her effective rate is reduced by the rate of the current. So, her rate when paddling upstream is (x - 2) km/hour.
When the canoeist is paddling downstream (with the current), her effective rate is increased by the rate of the current. So, her rate when paddling downstream is (x + 2) km/hour.
Given that the canoeist can paddle 12 km upstream and 12 km downstream in the same amount of time, we can set up the following equation:
Time upstream = Time downstream
12 / (x - 2) = 12 / (x + 2)
By cross-multiplying, we get:
12(x + 2) = 12(x - 2)
Simplifying, we have:
12x + 24 = 12x - 24
Subtracting 12x from both sides, we get:
24 = -24
This equation has no solution, which means that our initial assumption, x km/hour as the canoeist's rate in still water, is incorrect.
Therefore, there is no possible rate in still water that would satisfy the given conditions.