SOLUTION: Can somebody walk me through a step on this quadratic word problem? A small motorboat travels 13 mph in still water. It takes 2 hours longer to travel 60 miles going upstream

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Question 1042685: Can somebody walk me through a step on this quadratic word problem?

A small motorboat travels 13 mph in still water. It takes 2 hours longer to travel 60 miles going upstream than it does going downstream. Find the rate of the current.

I am at a point where I have:
60/13-x - 60/13+x =2
I know that I need to multiply both sides of the equation by the LCD and that it will turn into:
2x^2+20x-338=0
But I just cant seem to figure out HOW to actually do that. I think the denominators of 13-x/13+x are really throwing me.
Or if somebody has an easier way to get to the quadratic equation in this word problem I am all ears.
Thanks so much in advance!
Found 4 solutions by ikleyn, Boreal, ankor@dixie-net.com, stanbon:
Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
A small motorboat travels 13 mph in still water. It takes 2 hours longer to travel 60 miles going upstream
than it does going downstream. Find the rate of the current.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

60%2F%2813-x%29+-+60%2F%2813%2Bx%29 = 2

Multiply both sides of the equation by (13-x)*(13+x) 

60*(13+x) - 60*(13-x) = 2%2A%28169-x%5E2%29,

120x = 338+-+2x%5E2    <--- Your mistake was here.

2x^2 + 120x - 338 = 0,

x^2 + 60x - 169 = 0.   

Can you complete it from this point ?


Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
The speed upstream is 13-c, where c is the current.
The time it takes is 60/(13-c), as you have with units mi/mi/hr, with units hr.
The time it takes to go downstream is 60/(13+c), and that is 2 hours less than upstream
Therefore, 60/(13-c)-2=60/(13+c)
Your LCD is (13-c)(13+c)
60(13+c)-2(169-c^2)=60(13-c)
780 +60c-338+2c^2=780-60c
780 cancels,
2c^2+120c-338=0; Notice, this is 120c, not 20c.
divide by 2
c^2+60c-169=0
c=(1/2)(-60+/-sqrt (3600+676) sqrt (4276)=65.39
use positive root (1/2)5.39=2.695 mph
Upstream speed is 10.305 mph
Downstream speed is 15.695 mph
60/10.305=5.822 hours
60/15.695=3.822 hours
graph%28200%2C300%2C-10%2C10%2C-200%2C100%2Cx%5E2%2B60x-169%29

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
A small motorboat travels 13 mph in still water.
It takes 2 hours longer to travel 60 miles going upstream than it does going downstream.
Find the rate of the current.
:
You are definitely on the right track. Here it is step by step anyway.
:
let = the rate of the current
then
(13+r) = the effective speed down stream
and
(13-r) = the effective speed upstream
:
Write a time equation, time = dist/speed
Time upstream - time downstream = 2hr
60%2F%28%2813-r%29%29 - 60%2F%28%2813%2Br%29%29 = 2
multiply the equation by the common denominator
(13-r)(13+r)*60%2F%28%2813-4%29%29 - (13-r)(13+r)60%2F%28%2813%2Br%29%29 = 2(13-r)(13+r)
Cancel the denominators, FOIL on the right (difference of squares)
60(13+r) - 60(13-r) = 2(169-r^2)
distribute
780 + 60r - 780 + 60r = 338 - 2r^2
Combine like terms to form a quadratic equation on the left
2r^2 + 120r - 338 = 0
simplify, divide by 2
r^2 + 60r - 169 = 0
Using the quadratic formula; a=1; b=60; c=-169, I got solutions of of
r = - 69.7
and
r = 2.7 mph is the rate of the current (the positive solution is all we want)
:
:
:
See if this checks out by finding the actual time each way, Subtract and add the current from 13
60/10.3 = 5.825 hrs
60/15.7 = 3.822 hrs
--------------------
difference 2 hrs (discrepancy from me rounding off the original solutions)
:
:
Did this make sense to you. Let me know in the comment CK

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
A small motorboat travels 13 mph in still water. It takes 2 hours longer to travel 60 miles going upstream than it does going downstream. Find the rate of the current.
-----
I am at a point where I have:
60/(13-x) - 60/(13+x) =2
--------
60(13+x) - 60(13-x) = 2(169-x^2)
----------------------------
120x = 2*169 - 2x^2
2x^2+120x-2*169 = 0
---------------------------
x^2 + 60x - 169 = 0
----------------------
x = [-60 +- sqrt(3600-4*-169)]/(2)
-----
Positive solution::
x = 2.32 mph
--------------
Cheers,
Stan H.
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