A boat traveled 36 mi up a river in 3 hours, returning downstream, the boat took 2 hours. What is the boat's rate in still water and what is the rate of the river's current?
(Vb-Vr) * T1 = 36
(Vb-Vr)3 = 36
Eq1: Vb - Vr = 12
(Vb+Vr) * T2 = 36
(Vb+Vr)2 = 36
Eq2: Vb + Vr = 18
Add Eq1 and Eq2:
Vb - Vr = 12.
Vb + Vr = 18.
Sum: 2Vb = 30, Vb = 15 mi/h.
In Eq1, replace Vb with 15:
15 - Vr = 12, Vr = 3 mi/h.
To find the boat's rate in still water and the rate of the river's current, we can use the formula:
Boat's speed in still water + Current's speed = Speed downstream
Boat's speed in still water - Current's speed = Speed upstream
Let's assume the boat's speed in still water (the boat's rate) is represented by 'b', and the rate of the river's current is represented by 'c'.
Using the given information:
Speed downstream = 36 miles / 2 hours = 18 miles per hour
Speed upstream = 36 miles / 3 hours = 12 miles per hour
Now we can set up two equations based on the formulas mentioned earlier:
b + c = 18 -- Equation 1
b - c = 12 -- Equation 2
Now we can solve these two equations simultaneously to find the values of 'b' and 'c'.
To do this, we can add the two equations together:
(b + c) + (b - c) = 18 + 12
2b = 30
b = 30 / 2
b = 15
Now we have the boat's speed in still water, which is 15 miles per hour. We can substitute this value back into either Equation 1 or Equation 2 to find the rate of the river's current.
Using Equation 1:
15 + c = 18
c = 18 - 15
c = 3
Therefore, the boat's rate in still water is 15 miles per hour, and the rate of the river's current is 3 miles per hour.