Traveling upstream on the Mississippi River, a barge travels 56 mi in 7h. Downstream, it travels the same distance in 4h. Find the rate of the barge in still water and the rate of the current.
To find the rate of the barge in still water and the rate of the current, we can set up equations based on the given information.
Let's say the rate of the barge in still water is represented by "b", and the rate of the current is represented by "c".
When the barge is traveling upstream (against the current), its effective speed is reduced due to the opposing current. In this case, the effective speed is calculated as the difference between the rate of the barge in still water and the rate of the current:
Effective speed upstream = b - c
Similarly, when the barge is traveling downstream (with the current), its effective speed is increased due to the assistance of the current. In this case, the effective speed is calculated as the sum of the rate of the barge in still water and the rate of the current:
Effective speed downstream = b + c
Now, we can use these concepts to set up the equations based on the given information:
Upstream: distance = rate × time
56 miles = (b - c) × 7 hours
Downstream: distance = rate × time
56 miles = (b + c) × 4 hours
We now have a system of two equations with two variables. We can solve this system to find the values of "b" and "c".
Let's solve the system of equations:
From the first equation, we can express b in terms of c:
b = 56/7 + c
Now substitute this expression for b into the second equation:
56 = (56/7 + c + c) × 4
Simplifying the equation:
56 = 8(c + 7c)
56 = 8(8c)
56 = 64c
c = 56/64
c = 0.875
Now, substitute the value of c back into the expression for b:
b = 56/7 + 0.875
b = 8 + 0.875
b = 8.875
Therefore, the rate of the barge in still water is approximately 8.875 miles per hour, and the rate of the current is approximately 0.875 miles per hour.