SOLUTION: a motorboat travels 9 mph in still water. It takes 5 hours longer to travel 55 miles going upstream than it does going downstream. Find the rate of the current. Hint (x+9) is goin

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Question 813725: a motorboat travels 9 mph in still water. It takes 5 hours longer to travel 55 miles going upstream than it does going downstream. Find the rate of the current. Hint (x+9) is going downstream and (x-9) is going upstream. round to nearest tenth.

Found 3 solutions by mananth, josgarithmetic, MathTherapy:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
still water 9 mph
current - x mph
down stream = 9+x
upstream = 9-x
time upstream - time downstream = 5 hours
t=d/r
55/(9-x) -55/(9+x)=5
55(9+x)-55(9-x)=5(9^2-x^2)
495+55x-495+55x=405-5x^2
5x^2+110x-405=0
Find the roots of the equation by quadratic formula

a= 5 , b= 110 , c= -405

b^2-4ac= 12100 + 8100
b^2-4ac= 20200
%09sqrt%28%0920200%09%29=%09142.13%09
x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29
x1=%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29
x1=( -110 + 142.13 )/ 10
x1= 3.21
x2=%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29
x2=( -110 -142.13 ) / 10
x2= -25.21
Ignore negative value
current speed 3.2 mph


Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Direction__________speed____________time________________distance
UPstream___________x-9______________y+5_________________55
DOWNstream_________x+9______________y___________________55

x is as you have assigned, the speed of the boat in still water.
y is the time in hours to travel downstream the 55 miles.

Basic idea is Rate*Time=Distance. This gives two equations:
%28x-9%29%28y%2B5%29=55 and %28x%2B9%29%2Ay=55, neither linear.

xy-9y%2B5x-45=55 and xy%2B9y=55.
If we have one more equation, this might be easier. Trying to equate the distances, %28x-9%29%28y%2B5%29=y%28x%2B9%29
xy-9y%2B5x-40=xy%2B9y
-9y%2B5x-40=9y
-18y%2B5x=40
Continue solving this for x (or y if someone wants):
5x=40%2B18y
x=8%2B%2818%2F5%29y
Try substituting this into the simplified downstream equation:
xy%2B9y=55
%288%2B%2818%2F5%29y%29y%2B9y=55
8y%2B%2818%2F5%29y%5E2%2B9y-55=0
multiply both sides by 25,
25%2A17y%2B18y%5E2-25%2A55=0
18y%5E2%2B425y-1375=0

General Solution to Quadratic Equation:
y=%28-425%2B-+sqrt%28425%5E2-4%2A18%2A%281375%29%29%29%2F%282%2A18%29
y=%28-425%2Bsqrt%28279625%29%29%2F%282%2A18%29
279625=25*5*2237
y=%28-425%2B5%2Asqrt%285%2A2237%29%29%2F36
y=103.7958%2F36
highlight%28y=2.88%29 HOURS
This means that y+5 to go upstream was 7.88 hours.

Remember in this analysis, we found, x=8%2B%2818%2F5%29y ?
Now we can calculate the boat's speed in still water.
highlight%28x=8%2B%2818%2F5%29%2A%282.88%29=highlight%2818.4%29%29 miles per hour.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

a motorboat travels 9 mph in still water. It takes 5 hours longer to travel 55 miles going upstream than it does going downstream. Find the rate of the current. Hint (x+9) is going downstream and (x-9) is going upstream. round to nearest tenth.

Speed of current: 3.21267 ≈ highlight_green%283.2%29 mph

You can do the check!!

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