Question 919232
Let b = speed of boat in mph


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Upstream: You go slower since you are going against the current. The actual speed is b-5. The "-5" is due to the 5 mph current slowing you down.


{{{D = r*t}}}


{{{40 = (b-5)*t}}}


{{{40/(b-5) = t}}}


{{{t = 40/(b-5)}}}


It takes {{{40/(b-5)}}} hours to go upstream


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Upstream: You go faster since you are going with the current. The actual speed is b+5. The "+5" is due to the 5 mph current speeding you up.


{{{D = r*t}}}


{{{40 = (b+5)*t}}}


{{{40/(b+5) = t}}}


{{{t = 40/(b+5)}}}


It takes {{{40/(b+5)}}} hours to go downstream


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Add up the two time expressions {{{40/(b-5)}}} and {{{40/(b+5)}}}


{{{40/(b-5) + 40/(b+5)}}}


Set this sum equal to 5 because the two time values must add to 5 hours. This is the total round trip time.


{{{40/(b-5) + 40/(b+5) = 5}}}


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Now we solve for b.


{{{40/(b-5) + 40/(b+5) = 5}}}


{{{40(b+5) + 40(b-5) = 5(b-5)(b+5)}}} Multiply both sides by the LCD {{{(b-5)(b+5)}}} to clear out the fractions.


{{{40(b+5) + 40(b-5) = 5(b^2-25)}}}


{{{40b+200 + 40b-200 = 5b^2-125}}}


{{{80b = 5b^2-125}}}


{{{0 = 5b^2-125-80b}}}


{{{5b^2-80b-125 = 0}}}


{{{5(b^2-16b-25) = 0}}}


{{{b^2-16b-25 = 0}}}


Use the quadratic formula to get these solutions:


{{{b=8+sqrt(89)}}}, {{{b=8-sqrt(89)}}}


The solutions approximate to


{{{b=17.4339811320567}}}, {{{b = -1.4339811320566}}}


Toss out the negative answer because a negative speed makes no sense in this case.


The exact answer is {{{b=8+sqrt(89)}}}. The approximate answer is {{{b=17.4339811320567}}}


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Thanks,


Jim