Brian's boat has a top speed of 9 miles per hour in still water. While traveling on a river at top speed, he went 10 miles upstream in the same amount of time he went 20 miles downstream. Find the rate of the river current.
let the speed of the river current be x mph
speed going upstream = 9 -x mph
speed going downstream = 9 + x mph
10/(9-x) = 20/(9+x)
90 + 10x = 180 - 20x
30x = 90
x = 3
the speed of the river is 3 mph
To find the rate of the river current, we first need to set up equations based on the given information.
Let's assume the rate of the river current is "x" miles per hour.
When Brian is going upstream (against the current), the effective speed of the boat is decreased by the rate of the river current. So, his speed is 9 - x miles per hour.
Similarly, when Brian is going downstream (with the current), the effective speed of the boat is increased by the rate of the river current. So, his speed is 9 + x miles per hour.
Now, the time taken to go 10 miles upstream at a speed of 9 - x miles per hour is given as the same time taken to go 20 miles downstream at a speed of 9 + x miles per hour.
We can set up the following equation:
10 / (9 - x) = 20 / (9 + x)
To solve this equation, we can cross-multiply:
10(9 + x) = 20(9 - x)
90 + 10x = 180 - 20x
30x = 90
x = 90 / 30
x = 3
Therefore, the rate of the river current is 3 miles per hour.
To find the rate of the river current, we can use the formula:
Speed = Distance / Time
Let's assume the rate of the river current is "r" miles per hour.
When traveling upstream, Brian's effective speed is reduced since he is going against the current. Therefore, his speed is:
Effective speed upstream = Speed in still water - Rate of current
Thus, the effective speed upstream is (9 - r) miles per hour.
When traveling downstream, Brian's effective speed is increased since he is going with the current. Therefore, his speed is:
Effective speed downstream = Speed in still water + Rate of current
Thus, the effective speed downstream is (9 + r) miles per hour.
According to the problem statement, Brian took the same amount of time to travel both upstream and downstream. Let's call this time "t".
The distance traveled upstream is 10 miles, and the distance traveled downstream is 20 miles.
Using the formula Speed = Distance / Time, we can set up the following equations:
10 / t = (9 - r) -- Equation 1
20 / t = (9 + r) -- Equation 2
To solve for "r", let's solve the system of equations.
From Equation 1, we can rewrite it as:
10 = t * (9 - r)
Simplifying, we get:
10 = 9t - rt
Bringing the rt term to the left side, we have:
rt - 9t = -10
Factoring out "t", we get:
t(r - 9) = -10
Dividing both sides by (r - 9), we find:
t = -10 / (r - 9)
Now, let's substitute this expression for "t" into Equation 2:
20 / (-10 / (r - 9)) = (9 + r)
Simplifying, we get:
20 * (r - 9) = (-10) * (9 + r)
Expanding both sides, we have:
20r - 180 = -90 - 10r
Adding 10r to both sides, we get:
30r - 180 = -90
Adding 180 to both sides, we get:
30r = 90
Dividing both sides by 30, we find:
r = 3
Therefore, the rate of the river current is 3 miles per hour.