Question 113995
Let r=speed of the boat in still water


When the boat goes upstream, we subtract 3mph from r to get (the boat is slowed by 3mph)


{{{60=(r-3)t[1]}}}


When the boat goes downstream, we can add 3mph to r to get (the boat is sped up by 3mph)


{{{60=(r+3)t[2]}}}


Now since the distance is the same we can set the two equations equal to one another to get


{{{60/(r-3)=t[1]}}} Now solve for {{{t[1]}}} in the first equation



{{{60/(r+3)=t[2]}}} Now solve for {{{t[2]}}} in the second equation



Since the total time was 9 hr, this means {{{t[1]+t[2]=9}}}



{{{60/(r-3)+60/(r+3)=9}}} Plug in {{{t[1]=60/(r-3)}}} and {{{t[1]=60/(r+3)}}}



{{{(r-3)(r+3)(60/(r-3)+60/(r+3))=(r-3)(r+3)(9)}}} multiply by the LCD {{{(r-3)(r+3)}}}



{{{60(r+3)+60(r-3)=(r-3)(r+3)(9)}}} Distribute



{{{60(r+3)+60(r-3)=(r^2-9)(9)}}} Foil



{{{60r+180+60r-180=9r^2-81}}} Distribute again



{{{120r=9r^2-81}}} Combine like terms



{{{0=9r^2-81-120r}}} Subtract 120r from both sides



{{{0=9r^2-120r-81}}} Rearrange the terms



Now let's use the quadratic formula to solve for r:



Starting with the general quadratic


{{{ar^2+br+c=0}}}


the general solution using the quadratic equation is:


{{{r = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{9*r^2-120*r-81=0}}} ( notice {{{a=9}}}, {{{b=-120}}}, and {{{c=-81}}})





{{{r = (--120 +- sqrt( (-120)^2-4*9*-81 ))/(2*9)}}} Plug in a=9, b=-120, and c=-81




{{{r = (120 +- sqrt( (-120)^2-4*9*-81 ))/(2*9)}}} Negate -120 to get 120




{{{r = (120 +- sqrt( 14400-4*9*-81 ))/(2*9)}}} Square -120 to get 14400  (note: remember when you square -120, you must square the negative as well. This is because {{{(-120)^2=-120*-120=14400}}}.)




{{{r = (120 +- sqrt( 14400+2916 ))/(2*9)}}} Multiply {{{-4*-81*9}}} to get {{{2916}}}




{{{r = (120 +- sqrt( 17316 ))/(2*9)}}} Combine like terms in the radicand (everything under the square root)




{{{r = (120 +- 6*sqrt(481))/(2*9)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{r = (120 +- 6*sqrt(481))/18}}} Multiply 2 and 9 to get 18


So now the expression breaks down into two parts


{{{r = (120 + 6*sqrt(481))/18}}} or {{{r = (120 - 6*sqrt(481))/18}}}



Now break up the fraction



{{{r=+120/18+6*sqrt(481)/18}}} or {{{r=+120/18-6*sqrt(481)/18}}}



Simplify



{{{r=20 / 3+sqrt(481)/3}}} or {{{r=20 / 3-sqrt(481)/3}}}



So these expressions approximate to


{{{r=13.9772373998204}}} or {{{r=-0.643904066487102}}}



So our possible solutions are:

{{{r=13.9772373998204}}} or {{{r=-0.643904066487102}}}


However, we cannot have a negative speed, so our only solution is {{{r=13.9772373998204}}}



So the speed of the boat in still water is about 13.977 mph