A boat takes 2.0hr {\rm hr} to travel 26km {\rm km} down a river, then 6.0hr {\rm hr} to return.
(Vb+Vr)2 = 26 km
Eq1: 2Vb + 2Vr = 26
(Vb-Vr)6 = 26 km
Eq2: 6Vb - 6Vr = 26
Add Eq1 and Eq2:
2Vb + 2Vr = 26
6Vb - 6Vr = 26
Multiply Eq1 by 3:
6Vb + 6Vr = 78
6Vb - 6Vr = 26
Sum: 12Vb = 104
Vb = 8.6667 km/h = Velocity of the boat.
In Eq1, replace Vb with 8.6667:
2*8.6667 + 2Vr = 26
2Vr = 26-17.3333 = 8.6666 km/h.
Vr = 4.3333 m/s = Velocity of the river.
To solve this problem, we can break it down into two parts: the time it takes for the boat to travel downstream and the time it takes to travel upstream. Let's start with the downstream trip.
Given:
- Time taken to travel downstream: 2.0 hours
- Distance traveled downstream: 26 km
To find the boat's speed downstream, we can use the formula:
Speed = Distance / Time
Substituting the given values:
Speed downstream = 26 km / 2.0 hours = 13 km/hr
Now, let's move on to the upstream trip.
Given:
- Time taken to travel upstream: 6.0 hours
To find the boat's speed upstream, we can use the same formula:
Speed upstream = Distance / Time
Substituting the given values:
Speed upstream = 26 km / 6.0 hours = 4.33 km/hr
Therefore, the boat's speed downstream is 13 km/hr and its speed upstream is 4.33 km/hr.
To solve this problem, we need to find the speed of the boat and the speed of the river's current.
Let's assume the speed of the boat is B km/hr and the speed of the river's current is R km/hr.
When the boat is traveling downstream (down the river), it is aided by the river's current, so the effective speed of the boat is the sum of the boat's speed and the current's speed. Therefore, the effective speed is (B + R) km/hr. The distance traveled downstream is 26 km, and the time taken is 2 hours.
Using the formula: Speed = Distance / Time, we can write the equation:
(B + R) = 26 / 2
When the boat is traveling upstream (against the river's current), it is working against the current, so the effective speed is the difference between the boat's speed and the current's speed. Therefore, the effective speed is (B - R) km/hr. The distance traveled upstream is also 26 km, and the time taken is 6 hours.
Again, using the formula: Speed = Distance / Time, we can write the equation:
(B - R) = 26 / 6
Now we have a system of equations:
(B + R) = 26 / 2
(B - R) = 26 / 6
To solve this system of equations, we can use the method of substitution or elimination.
Let's use the method of substitution:
From the first equation, we can rearrange it to B = (26 / 2) - R = 13 - R.
Substitute B = 13 - R into the second equation:
(13 - R) - R = 26 / 6
13 - 2R = 26 / 6
Multiply both sides of the equation by 6 to get rid of the fraction:
78 - 12R = 26
Rearrange the equation to isolate R:
-12R = 26 - 78
-12R = -52
R = (-52) / (-12)
R = 4.33 km/hr (rounded to two decimal places)
Now, substitute the value of R back into one of the original equations to solve for B:
B + 4.33 = 13
B = 13 - 4.33
B = 8.67 km/hr (rounded to two decimal places)
Therefore, the speed of the boat is 8.67 km/hr and the speed of the river's current is 4.33 km/hr.