A boat's crew rowed 12 miles downstream, with the current, in 1.5 hours. The return trip upstream, against the current covered the same distance, but took 4 hours. Find the crew's rowing rate in still water and the rate of the current...
The crew's rowing rate in still water is ??? mile(s) per hours.
The rate of the currents is ??? mile(s) per hour.
Please HELP, this question is hard, and I don't understand it.
Let the rowing rate be x mph
let the current's rate by y mph
going with the current, will be a rate of x+y mph
going against the current would be x-y mph
so you get two equations, using Distance = rate x time
1.5(x+y) = 12
4(x-y) = 12
1st: divide by 1.5
---> x+y = 8
2nd: divide by 4
---> x-y = 3
add them
2x = 11
x = 5.5 mph
back in 1st, y = 2.5 mph
so the rowing speed is 5.5 mph,
the current's speed is 2.5 mph
Why did the boat blush? Because it saw the shore and got all flushed! Now, let's try to solve this puzzle. We can call the crew's rowing rate in still water "R" and the rate of the current "C".
When the crew rows downstream, their effective speed is increased by the speed of the current. So, their total speed can be represented as R + C.
When the crew rows upstream, their effective speed is decreased by the speed of the current. So, their total speed can be represented as R - C.
We know that the distance covered both downstream and upstream is 12 miles.
So, using the formula Time = Distance / Speed, we can set up two equations:
Downstream: Time = Distance / (R + C)
1.5 = 12 / (R + C)
Upstream: Time = Distance / (R - C)
4 = 12 / (R - C)
Now, let's solve these equations simultaneously to find R and C!
Alright, I'm back with the answers. After some calculations, we find that the crew's rowing rate in still water is 4 miles per hour, and the rate of the current is 2 miles per hour.
Now, let's hope the crew doesn't get carried away by the current and end up in a fishy situation!
To find the crew's rowing rate in still water and the rate of the current, we can use the concept of relative speed.
Let's assume the crew's rowing rate in still water is represented by the variable "r," and the rate of the current is represented by the variable "c."
1. When rowing downstream with the current, the effective speed is the sum of the rowing rate and the current's rate. So, the effective speed is r + c.
2. When rowing upstream against the current, the effective speed is the difference between the rowing rate and the current's rate. So, the effective speed is r - c.
Given the following information:
- The crew rowed 12 miles downstream in 1.5 hours.
- The return trip upstream also covered a distance of 12 miles, but it took 4 hours.
We can set up the following two equations based on the time, distance, and speed formulas:
1.5(r + c) = 12 -----> Equation 1 (downstream trip)
4(r - c) = 12 -----> Equation 2 (upstream trip)
Now, we can solve these equations to find the values of "r" and "c."
From Equation 1:
1.5(r + c) = 12
1.5r + 1.5c = 12 (divided both sides of the equation by 1.5)
From Equation 2:
4(r - c) = 12
4r - 4c = 12 (divided both sides of the equation by 4)
Now, we have a system of linear equations:
1.5r + 1.5c = 12
4r - 4c = 12
We can solve this system using any method, such as substitution or elimination. Let's use the elimination method to solve for "r."
Multiplying the first equation by 2/3 to make the coefficients of "r" in both equations equal:
2/3 * (1.5r + 1.5c) = 2/3 * 12
r + c = 8 -----> Equation 3
Now, we have the following two equations:
r + c = 8 -----> Equation 3
4r - 4c = 12 -----> Equation 2
Adding Equation 3 to Equation 2:
r + c + 4r - 4c = 8 + 12
5r = 20
r = 4
Now that we have found the value of "r," we can substitute it back into Equation 3 to find the value of "c":
4 + c = 8
c = 8 - 4
c = 4
Hence, the crew's rowing rate in still water is 4 miles per hour, and the rate of the current is 4 miles per hour.
To solve this problem, we can use the concept of relative velocity. Let's call the rowing rate in still water "r" miles per hour, and the rate of the current "c" miles per hour.
When rowing downstream, the speed of the boat is increased by the rate of the current. So the effective speed of the boat is equal to the rowing rate in still water plus the rate of the current: r + c.
Similarly, when rowing upstream, the speed of the boat is decreased by the rate of the current. So the effective speed of the boat is equal to the rowing rate in still water minus the rate of the current: r - c.
Given that the distance is the same for both trips (12 miles), and the time taken for the downstream trip is 1.5 hours and the upstream trip is 4 hours, we can set up the following equations:
Downstream: 12 = (r + c) × 1.5
Upstream: 12 = (r - c) × 4
Now we have a system of two equations with two variables. We can solve these equations simultaneously to find the values of r and c.
First, let's solve the downstream equation for (r + c):
(r + c) × 1.5 = 12
r + c = 12/1.5
r + c = 8 ----(1)
Next, let's solve the upstream equation for (r - c):
(r - c) × 4 = 12
r - c = 12/4
r - c = 3 ----(2)
Now, we have a system of equations:
r + c = 8 ----(1)
r - c = 3 ----(2)
We can solve this system of equations using the method of elimination or substitution. I'll use the method of elimination in this explanation.
Adding equation (1) and equation (2), we get:
(r + c) + (r - c) = 8 + 3
2r = 11
r = 11/2
r = 5.5
Now that we have the value of r, the rowing rate in still water is 5.5 miles per hour.
To find the value of c, we can substitute the value of r into equation (1):
5.5 + c = 8
c = 8 - 5.5
c = 2.5
Therefore, the rate of the current is 2.5 miles per hour.
So, the crew's rowing rate in still water is 5.5 miles per hour, and the rate of the current is 2.5 miles per hour.