A boat traveling downstream can travel 10 miles in 1/2 hour. The same distance traveling upstream will take 2 hours. What is the speed of the boat and the rate of the current?
speed of boat in still water ---- x mph
speed of current ---- y mph
(1/2)(x+y) = 10
x + y = 20
2(x-y) = 10
x - y = 5
add them
2x = 25
x = 12.5
then y = 7.5
state the conclusion.
To find the speed of the boat and the rate of the current, we can apply the concept of relative motion.
Let's assume the speed of the boat in still water is B mph, and the rate of the current is C mph.
When the boat is traveling downstream, the combined speed of the boat and the current adds up, so the effective speed is B + C mph. We are given that the boat can travel 10 miles in 1/2 hour downstream, so we can set up the equation:
(B + C) * (1/2) = 10
Simplifying the equation, we get:
B + C = 20
Similarly, when the boat is traveling upstream, the speed of the current subtracts from the speed of the boat, so the effective speed is B - C mph. We are given that the boat takes 2 hours to travel the same distance upstream, so we can set up the equation:
(B - C) * 2 = 10
Simplifying the equation, we get:
B - C = 5
Now, we have a system of equations:
B + C = 20
B - C = 5
We can solve this system of equations to find the values of B and C. Adding the two equations together, we get:
2B = 25
Dividing both sides by 2, we find:
B = 12.5
Substituting the value of B back into one of the equations, we can find the value of C:
12.5 + C = 20
C = 20 - 12.5
C = 7.5
Therefore, the speed of the boat in still water is 12.5 mph, and the rate of the current is 7.5 mph.