in still water, a paddle boat averages 10 miles per hour. It takes the boat the same amount of time to travel 36 mile downstream, with the current as 24 miles upstream, against the current. What is the rate of the water's current?
time = distance/speed
36/(10+c) = 24/(10-c)
c = 2
To find the rate of the water's current, we need to set up an equation based on the information given.
Let's assume that the speed of the paddle boat in still water is 'B' miles per hour and the rate of the water's current is 'C' miles per hour.
When the paddle boat is going downstream, the current helps in propelling the boat, so the effective speed is the sum of the speed of the paddle boat and the speed of the current. So, the speed downstream is B + C.
When the paddle boat is going upstream against the current, the current opposes the boat's movement, so the effective speed is the difference between the speed of the paddle boat and the speed of the current. So, the speed upstream is B - C.
Given that the paddle boat averages 10 miles per hour in still water, we can write the equation for downstream speed as:
B + C = 10
It takes the boat the same amount of time to travel 36 miles downstream, with the current as 24 miles upstream, against the current. We can use the formula:
Time = Distance / Speed
So, for traveling 36 miles downstream, the time taken is 36 / (B + C).
For traveling 24 miles upstream, the time taken is 24 / (B - C).
Since the times are the same, we can set up another equation:
36 / (B + C) = 24 / (B - C)
Now we have a system of equations:
B + C = 10
36 / (B + C) = 24 / (B - C)
We can solve this system of equations to find the values of B (speed of the boat in still water) and C (rate of the water's current).
Let's start by solving the first equation for B:
B = 10 - C
Now substitute this value of B into the second equation:
36 / (10 - C + C) = 24 / (10 - C)
Simplifying further:
36/10 = 24/(10 - C)
Now we can cross-multiply to get rid of the fraction:
36 * (10 - C) = 24 * 10
360 - 36C = 240
Rearranging the equation:
36C = 360 - 240
36C = 120
Dividing both sides by 36:
C = 120 / 36
C ≈ 3.33
Therefore, the rate of the water's current is approximately 3.33 miles per hour.