It takes Tom five hours to travel 20 miles downstream. It takes him eight hours to travel the same distance back up stream. How fast is the boat? How fast is the current?
speed of boat --- x mph
speed of current --- y mph
5(x+y) = 20
x+y = 4
8(x-y) = 20
x-y = 2.5
add them:
2x = 6.5
x = 3.25
then y = .75
state the conclusions.
v+c = 20/5 = 4
v-c = 20/8 = 2.5
----------------------add
2 v = 6.5
v = 3.25
c = 4 - 3.25 = .75
To find the speed of the boat and the current, we can use the concept of relative speed. Let's assume the speed of the boat is "b" and the speed of the current is "c".
When Tom is traveling downstream, the boat's speed and the current's speed add up, so the effective speed becomes: (b + c).
When Tom is traveling upstream, the boat's speed and the current's speed subtract, so the effective speed becomes: (b - c).
We are given that it takes Tom five hours to travel 20 miles downstream and eight hours to travel the same distance upstream.
Downstream:
Distance = Speed * Time
20 = (b + c) * 5
Upstream:
Distance = Speed * Time
20 = (b - c) * 8
We now have a system of two equations:
1) 20 = 5(b + c)
2) 20 = 8(b - c)
First, simplify equation 1:
20 = 5b + 5c
Next, simplify equation 2:
20 = 8b - 8c
Now we have a system of linear equations:
5b + 5c = 20
8b - 8c = 20
We can solve this system of equations using either substitution or elimination method.
Using elimination method, we'll multiply equation 1 by 8 and equation 2 by 5 to eliminate "c":
40b + 40c = 160
40b - 40c = 100
Adding the two equations together, we get:
80b = 260
Divide both sides by 80:
b = 260/80
b = 3.25 mph (speed of the boat)
Substituting the value of b into equation 1:
5(3.25) + 5c = 20
16.25 + 5c = 20
5c = 20 - 16.25
5c = 3.75
Divide both sides by 5:
c = 3.75/5
c = 0.75 mph (speed of the current)
Therefore, the speed of the boat is 3.25 mph and the speed of the current is 0.75 mph.