SOLUTION: Leaving from the same starting line at the same time, Tracy jogs along the riverbank at a constant speed, while Joe paddles a canoe at a constant speed in the river beside her. If

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Question 1127535: Leaving from the same starting line at the same time, Tracy jogs along the riverbank at a constant speed, while Joe paddles a canoe at a constant speed in the river beside her. If Joe was in still water, his speed would be the same as Tracy, but Joe is paddling against a 2 mile per hour current. If Tracy arrives at the end of her 6-mile jog 1 hour before Joe arrives at the same finish line, how fast was Tracy jogging?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
basic formula to use is rate * time = distance.
let R = rate and T = time and 6 = D

formula for tracy becomes RT = 6

formula for joe becomes (R - 2) * (T + 1) = 6

simplify the formula for joe to get RT + R - 2T - 2 = 6

since RT = 6 from the formula for tracy, then the formula for joe becomes 6 + R - 2T - 2 = 6 after you replace RT with 6.

subtract 6 from both sides of this equation to get 6 + R - 2T - 2 - 6 = 0

combine like terms to get R - 2T - 2 = 0

add 2T + 2 to both sides of the equation to get R - 2T - 2 + 2T + 2 = 2T + 2

combine like terms to get R = 2T + 2

raplace R with 2T + 2 in the formula for tracy to get (2T + 2) * T = 6

simplify and subtract 6 from both sides of the equation to get 2T^2 + 2T - 6 = 6 - 6

combine like terms to get 2T^2 + 2T - 6 = 0

divide both sides of the equation by 2 to get T^2 + T - 3 = 0

factor this quadratic equation to get T = -2.302775638 or T = 1.302775638.

since T can't be negative, then T has to be 1.302775638.

the formula for tracy becomes R * 1.302775638 = 6.

solve for R to get R = 6 / 1.302775638 = 4.605551275

confirm the original formulas are true with these values for T and R.

R * T = 6 becomes 4.605551275 * 1.302775638 = 6 which becomes 6 = 6, which is true.

(R - 2) * (T + 1) = 6 becomes 2.605551275 * 2.302775638 = 6 which becomes 6 = 6, which is also true.

the rate of speed of tracy is the same as the rate of speed of joe in still water, which is 4.605551275 miles per hour.

tracy jogs the 6 miles in 1.302775638 hours.

because joe is paddling against a 2 mile per hour current, than joe's net speed is 2.605551275 miles per hour and it takes him 2.302775638 hours to get to the 6 miles mark.

since tracy takes 1.302775638 hours to get to the 6 mile mark, she takes 1 hour less than joe.

the requirements of the problem are satisfied.

the solution is that tracy was jogging at 4.605551275 miles per hour.

you may round your answer as required.

the quadratic equation couldn't be factored so i had to resort to the use of the quadratic formula.

the quadratic formula is: 



            -b plus or minus sqrt(b^2-4ac)
     T =   -----------------------------------
                           2a


the equation to solve was T^2 + T - 3 = 0

a is the coefficient of the T^2 term which is equal to 1.
b is the coefficient of the T term which is equal to 1.
c is the constant term which is equal to -3.

plug these in the formula and you get: 



            -b plus or minus sqrt(b^2-4ac)
     T =   -----------------------------------
                           2a

becomes:

            -1 plus or minus sqrt(1^2-4*1*-3)
     T =   -----------------------------------
                           2*1

which becomes:

           -1 plus or minus sqrt(13))
     T =   -----------------------------------
                           2


this results in T = 1.302775638 or T = -2.302775638.

T has to be positive, so T = 1.302775638.

the rest of the problem was already solved above.




























Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Let x = "how fast is Tracy jogging".


Then Joe' speed in still water is the same x miles per hour, according to the condition,
and Joe's effective speed paddling against the current is (x-2) miles per hour.


Then the time equation is

 -  = 1 hour.


Thus the setup is done.


To solve equation, start multiplying both sides by x*(x-2).


6x - 6(x-2) = x*(x-2)


x^2 - 2x - 12 = 0


 =  =  = .


The positive root   = 4.605 mph is the solution. 


Check.   -  = 1.000   ! Correct !


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