A boat takes 3hr to travel 30 km down a river, then 5hr to return. How fast is the river flowing?
To determine the speed of the river, we need to consider the speed of the boat in still water and the difference between the boat's speed downstream and upstream.
Let's assume the speed of the boat in still water is "b" km/hr, and the speed of the river is "r" km/hr. When the boat is traveling downstream, it gets an additional speed boost from the river, so its effective speed is (b + r) km/hr. On the other hand, when the boat is traveling upstream, it has to overcome the flow of the river, resulting in an effective speed of (b - r) km/hr.
Given that the boat takes 3 hours to go downstream for a distance of 30 km, we can set up the equation:
30 = (b + r) * 3
Similarly, when the boat returns upstream over the same 30 km distance, it takes 5 hours:
30 = (b - r) * 5
Now we have a system of two equations. We can solve this system to find the values of "b" and "r" and determine the speed of the river.
Let's start by simplifying the equations:
Equation 1: 30 = 3b + 3r
Equation 2: 30 = 5b - 5r
Next, we can rewrite these equations to solve for "b":
Equation 1: 10 = b + r
Equation 2: 6 = b - r
Adding both equations together eliminates the "r" term:
10 + 6 = b + r + b - r
16 = 2b
Dividing both sides by 2 gives us the value of "b":
b = 16 / 2
b = 8 km/hr
Now that we know the speed of the boat in still water is 8 km/hr, we can substitute this value back into one of the original equations to find the speed of the river.
Let's use Equation 1:
30 = (b + r) * 3
30 = (8 + r) * 3
30 = 24 + 3r
3r = 30 - 24
3r = 6
r = 6 / 3
r = 2 km/hr
Therefore, the speed of the river is 2 km/hr.
To find the speed of the river, we can set up a system of equations based on the given information.
Let's assume the speed of the boat in still water is B km/h, and the speed of the river flow is R km/h.
When the boat is traveling downstream, it gets an additional boost from the river, so its effective speed is increased to (B + R) km/h.
Similarly, when the boat is traveling upstream, it has to fight against the current of the river, so its effective speed is decreased to (B - R) km/h.
The distance traveled downstream is 30 km, and the time taken is 3 hours. So, using the formula distance = speed × time, we can write the equation for downstream travel as:
30 = (B + R) × 3
The distance traveled upstream is also 30 km, but the time taken is 5 hours. So, the equation for upstream travel is:
30 = (B - R) × 5
We now have a system of two equations:
1) 30 = (B + R) × 3
2) 30 = (B - R) × 5
Let's solve these equations to find the values of B and R, which represent the speed of the boat in still water and the speed of the river flow, respectively.
Multiplying equation 1 by 5 and equation 2 by 3, we get:
5 × 30 = (B + R) × 3 × 5
3 × 30 = (B - R) × 5 × 3
Simplifying the equations, we have:
150 = 15B + 15R
90 = 15B - 15R
Adding these two equations, we eliminate the R terms:
240 = 30B
Dividing both sides by 30, we find:
B = 8
Now we can substitute this value back into one of the original equations to solve for R. Let's use equation 1:
30 = (B + R) × 3
30 = (8 + R) × 3
30 = 24 + 3R
3R = 30 - 24
3R = 6
R = 2
Therefore, the speed of the river flowing is 2 km/h.
Let V = stream velocity and v = boat velocity with respect to water. Both are in km/hr.
3 hr = 30/(V + v)
5 = 30/(V -v)
Solve those two equations in two unknowns for V and v.
5(V -v) = 3(V +v)
2V = 8v
v = V/4
3 = 30/(5V/4)= 120/5V
15V = 120
etc