the speed of the current in Puget Sound is 5 mph. A barge travels with the current 26 miles and returns in 10 and 2/3 hours. what is its speed in still water?
To find the speed of the barge in still water, we can use the concept of relative speed. In still water, the barge will move at a speed equal to its actual speed, as there is no current to either assist or hinder its movement.
Let's denote the speed of the barge in still water as "x" mph.
When the barge travels with the current, its effective speed is the sum of its speed in still water and the speed of the current. So, when going with the current, the barge will have a speed of (x + 5) mph.
We are given that when the barge travels 26 miles with the current, it takes a certain amount of time. To find this time, we can use the formula:
time = distance / speed
So, the time taken to travel 26 miles with the current is:
t1 = 26 / (x + 5)
When the barge travels against the current, its effective speed is the difference between its speed in still water and the speed of the current. So, when going against the current, the barge will have a speed of (x - 5) mph.
We are also given that when the barge returns the same 26 miles, but this time against the current, it takes 10 2/3 hours.
To find this time, we can first convert 10 2/3 hours to the fraction form: (32/3) hours.
Thus, the time taken to travel 26 miles against the current is:
t2 = 26 / (x - 5)
Now, we have two equations:
t1 = 26 / (x + 5)
t2 = 26 / (x - 5)
We need to solve these two equations to find the value of "x", which represents the barge's speed in still water.
Let's substitute the given values back into the equations:
26 / (x + 5) = (32/3)
26 / (x - 5) = (32/3)
To solve these equations, we can cross-multiply:
3 * 26 = (x + 5) * (32)
3 * 26 = (x - 5) * (32)
78 = 32x + 160
78 = 32x - 160
Combine like terms:
32x = -82
32x = 238
Divide each side by 32:
x = -82/32
x = 238/32
Simplifying:
x = -2.56
x = 7.44
Since speed can only be positive, we disregard the negative value:
x ≈ 7.44 mph
Therefore, the speed of the barge in still water is approximately 7.44 mph.