Question 171089: the current in a stream moves at a speed of 10 mph. A boat travels 29 miles upstreamand 29 miles downstream in a total time of 8 hours. What is the speed of the boat in still water
Found 2 solutions by checkley77, solver91311:
Answer by checkley77(12844)   (Show Source):
You can put this solution on YOUR website! 29/(B-10)+29/(B+10)=8
(29(B+10)+29(B-10))/(B-10)(B+10)=8
29B+290+29B-290=8(B^2-100)
58B=8B^2-800
8B^2-58B-800=0
USING THE QUADRATIC EQUATION: WE GET:
B=(58+-SQRT[-58^2-4*8*-800])/2*8
B=(58+-SQRT[3,364+25,600])/16
B=(58+-SQRT28,964)/16
B=(58+-170.18813)/16
B=(58+170.18813)/16
B=228.18813/16
B=14.261758 mph IS THE SPEED OF THE BOAT.
PROOF:
29/(14.261758-10)+29/(14.261758+10)=8
29/4.261758+29/24.261758=8
6.805+1.195
8=8
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website! The first thing I would do would be to make complaint about being given this computational horror, but here goes anyway.
Assume  is the rate of the boat in still water. The upstream rate is the speed of the boat in still water minus the current speed:  . The downstream rate is the rate in still water PLUS the current speed:  .
Assume  is the time it takes to go upstream 29 miles, then since the total time of the trip is 8 hours, the time to go downstream must be  .
Using the basic relationship  , written as  , the upstream trip can be expressed as:
Likewise, the downstream trip can be expressed as 
Isolating in both equations produces:
 and 
This gives us two different expressions for the elapsed time of the upstream trip in terms of the rate of the boat in still water. We can set these two expressions equal to each other:
Application of the common denominator  , setting the resulting numerators equal to each other and collecting like terms results in the following: (Intermediate steps left as an exercise for the student)
Applying the quadratic formula:
 (this root less than zero, so exclude as extraneous}
 cannot be simplified further because 7241 has no perfect square factors and therefore is the exact answer. A little calculator work results in a numerical approximation of 14.26 hours elapsed time for the upstream trip. This answer should properly be rounded to the nearest whole hour because your input values of distance, time, and current speed were all expressed in whole numbers and it is incorrect (unless greater precision is specified in the problem statement) to express a result of a computation involving measurements to a greater precision than the least precise input measurement.
Calculating the upstream rate, downstream rate, and time for each part of the trip to verify the sum is approximately equal to 8 hours, thereby verifying the answer, is left as an exercise for the student. Enjoy.
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