The speed of the current in a stream is 4 mi/hr. It takes a canoeist 96 minutes longer to paddle 13 miles upstream than to paddle the same distance downstream. What is the canoeist's rate in still water?
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To find the canoeist's rate in still water, we first need to analyze the situation and set up some equations.
Let's assume that the canoeist's rate in still water is represented by "C" in miles per hour (mi/hr).
When paddling downstream, the canoeist's speed is increased by the speed of the current, which is 4 mi/hr. Therefore, the canoeist's effective speed downstream can be expressed as "C + 4" mi/hr.
Conversely, when paddling upstream, the canoeist's speed is decreased by the speed of the current, so the effective speed upstream is "C - 4" mi/hr.
We are given that it takes the canoeist 96 minutes longer to paddle 13 miles upstream than to paddle the same distance downstream. This information can be expressed as an equation:
Time upstream - Time downstream = 96 minutes
To convert the time difference to hours, we divide 96 minutes by 60 (since there are 60 minutes in an hour):
96 minutes / 60 minutes per hour = 1.6 hours
Therefore, the equation becomes:
(Time upstream) - (Time downstream) = 1.6 hours
Now we need to calculate the time it takes for the canoeist to travel 13 miles downstream and upstream.
Time = Distance / Speed
For the downstream journey, the time is:
Time downstream = 13 miles / (C + 4) mi/hr
For the upstream journey, the time is:
Time upstream = 13 miles / (C - 4) mi/hr
Now we substitute these values into the equation:
(13 miles / (C - 4) mi/hr) - (13 miles / (C + 4) mi/hr) = 1.6 hours
We can simplify the equation by multiplying both sides by (C - 4)(C + 4) to eliminate the denominators:
13(C + 4) - 13(C - 4) = 1.6(C - 4)(C + 4).
Expanding both sides of the equation:
13C + 52 - 13C + 52 = 1.6(C^2 - 16)
Simplifying and collecting like terms:
52 + 52 = 1.6C^2 - 25.6
104 = 1.6C^2 - 25.6
Rearranging the equation:
1.6C^2 = 129.6
Dividing both sides by 1.6:
C^2 = 81
Taking the square root of both sides:
C = ±9
Since we are looking for a rate in still water, a negative rate wouldn't make sense in this context. Therefore, the canoeist's rate in still water is 9 mi/hr.