SOLUTION: 3) During the first part of a trip, a canoeist travels 82 miles at a certain speed. The canoeist travels 19 miles on the second part of the trip at a speed 5 mph slower. The total

Algebra ->  Rational-functions -> SOLUTION: 3) During the first part of a trip, a canoeist travels 82 miles at a certain speed. The canoeist travels 19 miles on the second part of the trip at a speed 5 mph slower. The total       Log On


   

Question 129829: 3) During the first part of a trip, a canoeist travels 82 miles at a certain speed. The canoeist travels 19 miles on the second part of the trip at a speed 5 mph slower. The total time for the trip is 3 hrs. What was the speed on each part of the trip?
The speed on the first part of the trip was _______ mph.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's use the formula d=r*t

Since "a canoeist travels 82 miles at a certain speed", this means that our first equation is

82=rt%5B1%5D Notice how I plugged in d=82 and t=t%5B1%5D


Also, since the "canoeist travels 19 miles on the second part of the trip at a speed 5 mph slower", the second equation is

19=%28r-5%29t%5B2%5D Notice how I plugged in d=19 and t=t%5B2%5D. Also, I replaced r with r-5 (since the canoeist traveled 5 mph slower)


Finally, because the "total time for the trip is 3 hrs", the third equation is


t%5B1%5D%2Bt%5B2%5D=3



So we have the three equations


82=rt%5B1%5D

19=%28r-5%29t%5B2%5D

t%5B1%5D%2Bt%5B2%5D=3


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82=rt%5B1%5D Start with the first equation


82%2Fr=t%5B1%5D Divide both sides by r to isolate t%5B1%5D


t%5B1%5D=82%2Fr Rearrange the equation



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19=%28r-5%29t%5B2%5D Start with the second equation


19%2F%28r-5%29=t%5B2%5D Divide both sides by r-5 to isolate t%5B2%5D


t%5B2%5D=19%2F%28r-5%29 Rearrange the equation



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Now let's focus on the third equation t%5B1%5D%2Bt%5B2%5D=3

82%2Fr%2B19%2F%28r-5%29=3 Plug in t%5B1%5D=82%2Fr and t%5B2%5D=19%2F%28r-5%29 into the third equation



r%28r-5%29%2882%2Fr%2B19%2F%28r-5%29%29=r%28r-5%29%283%29 Multiply both sides by the LCD r%28r-5%29



82%28r-5%29%2B19r=3r%28r-5%29 Distribute and multiply


82r-410%2B19r=3r%5E2-15r Distribute again



82r-410%2B19r-3r%5E2%2B15r=0 Subtract 3r^2 from both sides. Add 15r to both sides.


-3r%5E2%2B116r-410=0 Combine like terms


So after using the quadratic formula or a calculator, we get the solutions

r=3.93492 or r=34.73175


However, since you cannot take 5 mph away from 3.93 mph (you'll get a negative answer if you do), this means that our only answer is

r=34.73175


So the speed on the first part of the trip was about 34.73 mph