Question 1151632: A boat’s speed in still in water is 4km/h.The boat cruises from A to B along a river flowing at an average speed of X km/h in the direction A to B . If the distance AB is 5 km and the boat takes 2hr more on its return journey,determine x. Hence,find the total time taken for the whole journey.
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let 4 = the rate of the boat in still water.
let R = the rate of the river.
let distance = 5
rate * time = distance.
with the current, the rate becomes (4 + R)
against the current, the rate becomes (4 - R)
let T = the time it takes going with the current.
then T + 2 = the time it takes coming back against the current.
the formulas are:
(4 + R) * T = 5
(4 - R) * (T + 2) = 5
simplify these equations to get:
4T + RT = 5
4T + 8 - RT - 2R = 5
since they both equal 5, set them equal to each other to get:
4T + RT = 4T + 8 - RT - 2R
the 4T cancels out and you get RT = 8 - RT - 2R
subtract the right side of the equation from both sides of the equation to get:
RT - 8 + RT + 2R = 0
combine like terms to get:
2RT + 2R - 8 = 0
factor the first original equation of 4T + RT = 5 to get:
T * (R + 4) = 5
divide both sides of this equation by (R + 4) to get:
T = 5 / (R + 4)
replace T in 2RT + 2R - 8 = 0 to get:
2R * 5 / (R + 4) + 2R - 8 = 0
simplify to get:
10R / (R + 4) + 2R - 8 = 0
multiply both sides of this equation by (R + 4) to get:
10R + 2R * (R + 4) - 8 * (R + 4) = 0
simplify to get:
10R + 2R^2 + 8R - 8R - 32 = 0
combine like terms to get:
2R^2 + 10R - 32 = 0
divide both sides of this equation by 2 to get:
R^2 + 5R - 16 = 0
factor this quadratic to get:
R = 2.2169905660283 or R = -7.2169905660283
R has to be positive, so R = 2.2169905660283
your original equations are:
(4 + R) * T = 5
(4 - R) * (T + 2) = 5
when R = 2.2169905660283, the first equation becomes:
(4 + 2.2169905660283) * T = 5
solve for T to get:
T = 5 / 6.2169905660283 = .8042476415
your two original equations are, once again:
(4 + R) * T = 5
(4 - R) * (T + 2) = 5
when R = 2.2169905660283 and T = .8042476415, these equations become:
(4 + 2.2169905660283) * .8042476415 = 5
(4 - 2.2169905660283) * (.8042476415 + 2) = 5
use your calculator to confirm these equations are true.
i confirmed using my calculator.
your solution is that the total time taken for the whole journey is .8042476415 + .8042476415 + 2 = 3.608495283 hours.
it's not exactly a clean answer, but it does check out, assuming my original equations are correct.
i believe they are:
i used the following quadratic equation solver to find the roots of the quadratic.
https://www.mathsisfun.com/quadratic-equation-solver.html
this would have been a huge chore otherwise.
i didn't want to go through the large manual effort only to find that the solution was not good.
it turned out that the answer was good, even if it was messy.
the basic concepts is:
when you're going with the current, the formula becomes (B + R) * T = 5
when you're going against the current, the formula becomes (B - R) * (T + 2) = 5
Answer by MathTherapy(10552)  (Show Source):
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