A swimmer going downstream takes 1hour, 20 minutes to travel a certain distance. If takes the swimmer 4 hours to make to return trip against the current. If the river flows at the rate of 1.5mph. Find the rate of the swimmer in still water and the distance traveled one way.
let v be the swimmer velocity in still waters.
1.333 hr=distanceoneway/(v-1.5)
4hr=distance/(v+1.5)
in each equation, solve for distance, and since the distances are the same, then set those equations equal.
1.333(v-1.5)=4(V+1.5)
solve for v.
To solve this problem, let's denote the rate of the swimmer in still water as "x" mph and the distance traveled one way as "d" miles.
When the swimmer is going downstream, the river's current aids their movement, so their effective speed is increased by the rate of the current. Therefore, the swimmer's speed going downstream is (x + 1.5) mph.
We know that it takes the swimmer 1 hour and 20 minutes to go downstream, which is equivalent to 1.33 hours. So, the equation for the downstream journey can be written as:
d = (x + 1.5) * 1.33
On the other hand, when the swimmer is going upstream against the current, the river's current hinders their movement. Therefore, the swimmer's speed going upstream is (x - 1.5) mph.
We are given that it takes the swimmer 4 hours to make the return trip upstream. So, the equation for the upstream journey can be written as:
d = (x - 1.5) * 4
Now, we have a system of two equations:
d = (x + 1.5) * 1.33
d = (x - 1.5) * 4
To solve this system, we can equate the two equations and solve for x:
(x + 1.5) * 1.33 = (x - 1.5) * 4
Let's simplify the equation:
1.33x + 1.99 = 4x - 6
Move all the terms with "x" to one side:
4x - 1.33x = 1.99 + 6
Combine like terms:
2.67x = 7.99
Divide both sides by 2.67 to solve for x:
x = 7.99 / 2.67
x ≈ 2.993 = 3 (rounded to the nearest whole number)
Therefore, the swimmer's rate in still water is approximately 3 mph.
To find the distance traveled one way, we can substitute the value of x into either of the original equations. Let's use the equation for the downstream journey:
d = (3 + 1.5) * 1.33
Simplify the equation:
d ≈ 4.995 ≈ 5 (rounded to the nearest mile)
Therefore, the distance traveled one way is approximately 5 miles.