The time required for a trip 108 miles downstream on a steamer is 3 hours less than the time required for the upstream trip. A boat whose rate is 6 miles per hour less than that of the steamer required 9 hours more for the upstream trip than for the downstream trip. Find the rate of the stream.
To solve this problem, we need to set up a system of equations based on the given information.
Let's assume the rate of the stream is "r" miles per hour.
Let's also assume the rate of the steamer (boat going downstream) is "s" miles per hour, and the rate of the boat (boat going upstream) is "b" miles per hour.
Since the time required for the downstream trip is 3 hours less than the upstream trip, we can write the equation:
108 / (s + r) = 108 / (b - r) + 3
Similarly, since the boat required 9 hours more for the upstream trip than the downstream trip, we can write another equation:
108 / (b - r) = 108 / (s + r) + 9
Now, let's solve this system of equations:
First, let's simplify the equations by cross-multiplying:
1. 108(b - r) = (s + r)(108 + 3(s + r))
2. 108(s + r) + 9(b - r) = 108(b - r)
Expanding the expressions:
1. 108b - 108r = 108s +108r + 3s^2 + 3r^2 + 6sr
2. 108s + 108r + 9b - 9r = 108b - 108r
Combining like terms:
1. 3s^2 + 3r^2 + 6sr = 108b - 108s
2. 108b - 108s + 9b = 0
Simplifying further:
1. 3s^2 + 3r^2 + 6sr - 108b + 108s = 0
2. 117b - 108s = 0
Now, we can solve this system of equations using substitution or elimination.
From equation 2, we can solve for b in terms of s:
117b = 108s
b = 108s / 117
b = 12s / 13
Now substitute this value of b in equation 1:
3s^2 + 3r^2 + 6sr - 108(12s / 13) + 108s = 0
Simplifying further:
3s^2 + 3r^2 + 6sr - 864s / 13 + 108s = 0
Multiplying everything by 13 to eliminate the fraction:
39s^2 + 39r^2 + 78sr - 864s + 1404s = 0
Simplifying further:
39s^2 + 39r^2 + 78sr + 540s = 0
Factoring out 39:
39(s^2 + r^2 + 2sr + 14s) = 0
We know that s^2 + r^2 is always positive, so the only way for this equation to hold true is if:
s + r = 0
Since we are interested in finding the value of the stream rate "r", we can substitute this result back into one of the original equations we had:
s - r = 0
Simplifying:
s = r
Now, we can substitute this value of s into equation 2:
108b - 108s + 9b = 0
Substituting s = r:
108b - 108r + 9b = 0
Simplifying:
117b = 108r
b = 108r / 117
b = 12r / 13
Since the rate of the boat is 6 miles per hour less than that of the steamer, we can write:
b = s - 6
Substituting the values we found:
12r / 13 = r - 6
Simplifying:
12r = 13r - 78
Rearranging the terms:
r = 78
Therefore, the rate of the stream is 78 miles per hour.