SOLUTION: a motorboat heads upstream for a distance of 24 miles on the Illinois River, whose current is running at 3mph. the trip up and back takes a total of 6 hours. assuming that the boat

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Question 167652: a motorboat heads upstream for a distance of 24 miles on the Illinois River, whose current is running at 3mph. the trip up and back takes a total of 6 hours. assuming that the boat maintained a constant speed relative to the water, what is this constant speed? -- i've never been great at calculus word problems, thank u for all the help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Downstream:

d=rt Start with the distance-rate-time formula


24=rt Plug in d=24


24=%28r%2B3%29t Replace r with r%2B3 (since the speed of the river sped up the boat 3 mph)


t=24%2F%28r%2B3%29 Divide both sides by r%2B3 and rearrange to isolate "t"


So the time it took to go downstream was t=24%2F%28r%2B3%29

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Upstream:

d=rt Start with the distance-rate-time formula


24=rt Plug in d=24


24=%28r-3%29t Replace r with r-3 (since the speed of the river slowed the boat 3 mph)


t=24%2F%28r-3%29 Divide both sides by r-3 and rearrange to isolate "t"


So the time it took to go upstream was t=24%2F%28r-3%29



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Now simply add the two times and set them equal to the total time (6 hours) to get


Downstream Time + Upstream Time = Total Time

24%2F%28r%2B3%29%2B24%2F%28r-3%29=6


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24%2F%28r%2B3%29%2B24%2F%28r-3%29=6 Start with the given equation


Multiply EVERY term by the LCD %28r%2B3%29%28r-3%29 to clear the fractions


24%28r-3%29%2B24%28r%2B3%29=6%28r%2B3%29%28r-3%29 Multiply and simplify


24%28r-3%29%2B24%28r%2B3%29=6%28r%5E2-9%29 FOIL


24r-72%2B24r%2B72=6r%5E2-54 Distribute


48r=6r%5E2-54 Combine like terms.


0=6r%5E2-54-48r Subtract 48r from both sides


0=6r%5E2-48r-54 Rearrange the terms.


Notice we have a quadratic equation in the form of ar%5E2%2Bbr%2Bc where a=6, b=-48, and c=-54


Let's use the quadratic formula to solve for r


r+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


r+=+%28-%28-48%29+%2B-+sqrt%28+%28-48%29%5E2-4%286%29%28-54%29+%29%29%2F%282%286%29%29 Plug in a=6, b=-48, and c=-54


r+=+%2848+%2B-+sqrt%28+%28-48%29%5E2-4%286%29%28-54%29+%29%29%2F%282%286%29%29 Negate -48 to get 48.


r+=+%2848+%2B-+sqrt%28+2304-4%286%29%28-54%29+%29%29%2F%282%286%29%29 Square -48 to get 2304.


r+=+%2848+%2B-+sqrt%28+2304--1296+%29%29%2F%282%286%29%29 Multiply 4%286%29%28-54%29 to get -1296


r+=+%2848+%2B-+sqrt%28+2304%2B1296+%29%29%2F%282%286%29%29 Rewrite sqrt%282304--1296%29 as sqrt%282304%2B1296%29


r+=+%2848+%2B-+sqrt%28+3600+%29%29%2F%282%286%29%29 Add 2304 to 1296 to get 3600


r+=+%2848+%2B-+sqrt%28+3600+%29%29%2F%2812%29 Multiply 2 and 6 to get 12.


r+=+%2848+%2B-+60%29%2F%2812%29 Take the square root of 3600 to get 60.


r+=+%2848+%2B+60%29%2F%2812%29 or r+=+%2848+-+60%29%2F%2812%29 Break up the expression.


r+=+%28108%29%2F%2812%29 or r+=++%28-12%29%2F%2812%29 Combine like terms.


r+=+9 or r+=+-1 Simplify.


So the possible answers are r+=+9 or r+=+-1

However, a negative speed doesn't make much sense. So the only answer is r+=+9


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Answer:

So the solution is r+=+9 which means that the boat was going 9 mph relative to the water.


In other words, the speed of the boat in still water is 9 mph.