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?
To solve this problem, let's assume the canoeist's rate in still water is "r" miles per hour.
When paddling downstream, the speed of the current aids the canoeist in their movement. So, the effective speed becomes "r + 4" miles per hour.
When paddling upstream, the speed of the current opposes the canoeist's movement, so the effective speed becomes "r - 4" miles per hour.
We know that the canoeist takes 96 minutes longer to paddle upstream than to paddle downstream. Since there are 60 minutes in an hour, we can convert this to hours by dividing 96 minutes by 60, which gives us 1.6 hours.
Now, we can set up the equation based on the time, distance, and speed:
Time downstream = Time upstream + 1.6 hours
Distance = Rate × Time
For the downstream trip:
13 = (r + 4) × t1 ... where t1 is the time taken downstream in hours
For the upstream trip:
13 = (r - 4) × (t1 + 1.6) ... where t1 + 1.6 is the time taken upstream in hours
We can rearrange the second equation to isolate t1:
13 = (r - 4) × t1 + (r - 4) × 1.6
13 = (r - 4) × t1 + 1.6r - 6.4
13 = rt1 - 4t1 + 1.6r - 6.4
11.4 = rt1 - 4t1 + 1.6r
Now we have two equations:
13 = (r + 4) × t1
11.4 = rt1 - 4t1 + 1.6r
We can solve this system of equations to find the value of r, which represents the canoeist's rate in still water.
By solving these equations, we find that the canoeist's rate in still water is approximately 8.8 miles per hour.