Ashley can paddle her kayak 6 miles per hour in still water. It takes her as long to paddle 5 miles up stream as 10 miles downstream in the Maumee River. Determine the river’s current.
2 4 6 or 8?
River’s current—r:
5/(6 – r) = 10/(6 + r)
1/(6 – r) = 2/(6 + r)
6 + r = 12 – 2r
3r = 6
r = 2
Answer: 2 mph is the river’s current.
Proof (equal no. of time to paddle upstream & downstream):
5 mi/(6 mph – 2mph) = 10 mi/(6 mph + 2 mph)
5 mi/4 mph = 10 mi/8 mph
5/4 hr = 10/8 hr
5/4 hr = 5/4 hr
River’s current—r:
5/(6 – r) = 10/(6 + r)
1/(6 – r) = 2/(6 + r)
6 + r = 12 – 2r
3r = 6
r = 2
Answer: 2 mph is the river’s current.
Proof (equal no. of time to paddle upstream & downstream):
5 mi/(6 mph – 2mph) = 10 mi/(6 mph + 2 mph)
5 mi/4 mph = 10 mi/8 mph
5/4 hr = 10/8 hr
5/4 hr = 5/4 hr
References :
We can rule out 6 and 8 right away. If the current is 6, she goes 6-6=0 miles/hour upstream. If the current is 8, she goes downstream at 2 miles/hour even when she’s trying to go upstream.
Let x = the speed of the current
Then 6+x = the speed at which she travels downstream
And 6-x = the speed at which she travels upstream
Distance/Speed = Time
10/(6+x) = 5/(6-x)
Flip ‘em upside down to get:
(6+x)/10 = (6-x)/5
Multiply both sides by 10:
6+x = 12-2x
3x=6
x=2 miles per hour
References :
You know that she can paddle 6 mph in still water, which means that the water has a current of 0 mph. Let’s say that the time it takes her to travel upstream is t, and that the river’s current is c. When she is going upstream the current is in the opposite, or negative, direction. When she is going downstream the current is going in the same, or positive, direction. This means that her total speed going upstream is:
6 – c
And that her total speed going downstream is:
6 + c
You can see from the units of speed (or velocity) that it is a change in distance over time (mph, kph, mps, etc.). In this question the distances are 5 miles and 10 miles. So her speed going upstream can also be represented by:
5/t
And her speed going downstream can also be represented by:
10/t
This means that, from going upstream, we have:
6 – c = 5/t
And from going downstream we have:
6 + c = 10/t
Solving for t we have two equations:
(6-c)t = 5 -> t = 5/(6-c)
(6+c)t = 10 -> t = 10/(6+c)
These two equations are both equal to t, so now we have:
5/(6-c) = 10/(6+c)
Since (1/2)*(2/1) = (1/2)*2 = 1, we know that [1/(6-c)]*[(6-c)/1] = 1 and so forth:
(6-c)*[5/(6-c)] = [10/(6+c)]*(6-c)
5 = (10*(6-c))/(6+c)
(6+c)*5 = [(10*(6-c))/(6+c)]*(6+c)
(6+c)*5 = 10*(6-c)
30+5c = 60 – 10c
30 – 30 + 5c + 10c = 60 – 30 – 10c +10c
0 + 15c = 30 + 0
(15c)/15 = 30/15
c = 2
So the river’s current is 2 mph.
References :
B.S. degrees in Physics and Mathematics