in still water, a speed boat travels 5 times faster than the current of the river. If the speed boat can travel 48 miles upstream and then back in 5 hours, find the rate of the current.
If the current speed is c, then the boat's speed is 5c.
Since time = distance/speed,
48/(5c-c) + 48/(5c+c) = 5
12/c + 8/c = 5
...
To solve this problem, let's assume that the speed of the current is represented by "C" and the speed of the speed boat is represented by "S".
First, let's find the speed of the speed boat relative to the still water:
Since the speed boat travels 5 times faster than the current, we can write the equation:
S = 5C
Next, we can calculate the time it takes for the speed boat to travel upstream and downstream.
When the speed boat travels upstream, it moves against the current, reducing its effective speed. When it travels downstream, it moves with the current, increasing its effective speed.
Let's assume the speed of the boat relative to the still water is S.
When traveling upstream, the effective speed is reduced by the current, so its speed relative to the ground is (S - C).
When traveling downstream, the effective speed is increased by the current, so its speed relative to the ground is (S + C).
The distance traveled is the same for both the upstream and downstream journeys and is equal to 48 miles.
Now, let's calculate the time it takes for the speed boat to travel upstream and downstream:
Time to travel upstream: distance / relative speed = 48 / (S - C)
Time to travel downstream: distance / relative speed = 48 / (S + C)
According to the problem, the total time for the round trip is 5 hours. So we can write the equation:
48 / (S - C) + 48 / (S + C) = 5
Now, we have two equations:
S = 5C (Equation 1)
48 / (S - C) + 48 / (S + C) = 5 (Equation 2)
We can substitute Equation 1 into Equation 2 to solve for C:
48 / (5C - C) + 48 / (5C + C) = 5
48 / 4C + 48 / 6C = 5
12 / C + 8 / C = 5
(12 + 8) / C = 5
20 / C = 5
20 = 5C
C = 20 / 5
C = 4
Therefore, the rate of the current is 4 mph.