SOLUTION: Ineed your help once again I try own my own and my answer are looking craazt help is needed. Thank you so much Steve traveled 200 miles at a certain speed. Had he gone 10mph fa

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Question 172638This question is from textbook
: Ineed your help once again I try own my own and my answer are looking craazt help is needed. Thank you so much
Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle
can you show me this one too.
The Hudson River flows at a rate of 3 miles per hour. A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours. What is the speed of the boat in still water?
This question is from textbook

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
I set up the equations you solve
:
d=rt........in both instances distance is equal
lets call the rate of the slow trip r and time t
lets call the rate of the fast trip r+10 and time(t-1)
:
so 200=rt...eq 1
...200=(r+10)(t-1)..eq 2
:change eq 1 to t=200/r and substitute it into eq 2 and solve for r...this will involves a quadratic equation again
:
200=%28r%2B10%29%28%28200%2Fr%29-1%29
:
200=%28r%2B10%29%28%28200-r%29%2Fr%29
:
200r=%28r%2B10%29%28200-r%29
:
200r=200r-r%5E2%2B2000-10r
:
r%5E2%2B10r-2000=0
throw out negative valuesystem%28r=40%2Cr=-50%29
so highlight%28r=40%29mph and highlight%28r%2B10%29=50mph
:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ar%5E2%2Bbr%2Bc=0 (in our case 1r%5E2%2B10r%2B-2000+=+0) has the following solutons:

r%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2810%29%5E2-4%2A1%2A-2000=8100.

Discriminant d=8100 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-10%2B-sqrt%28+8100+%29%29%2F2%5Ca.

r%5B1%5D+=+%28-%2810%29%2Bsqrt%28+8100+%29%29%2F2%5C1+=+40
r%5B2%5D+=+%28-%2810%29-sqrt%28+8100+%29%29%2F2%5C1+=+-50

Quadratic expression 1r%5E2%2B10r%2B-2000 can be factored:
1r%5E2%2B10r%2B-2000+=+1%28r-40%29%2A%28r--50%29
Again, the answer is: 40, -50. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B10%2Ax%2B-2000+%29


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:
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Let s= speed of the boat in still water
The current =3 mi/hr
we know that d=rt or d/r=t for this problem we must break this up into 2 parts
(distance upriver)/(rate going upriver)
+ (distance downriver)/(rate going downriver) = 9 hrs
:the we know the distance is 60 each way
the rate going up stream is 60/s-3
the rate going down stream is 60/s+3
:
so we have 60%2F%28s-3%29%2B60%2F%28s%2B3%29=9again a quadratic equation
:
:
multiply each term by (s+3)(s-3)
:
60%28s%2B3%29%2B60%28s-3%29=9%28s%2B3%29%28s-3%29
60s%2B180%2B60s-180=9%28s%5E2-9%29
120s=9s%5E2-81
9s%5E2-120s-81=0
:
drop the negative valuesystem%28s=13.98%2Cs=-.64%29
:
so highlight%28s=14%29 mph

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation as%5E2%2Bbs%2Bc=0 (in our case 9s%5E2%2B-120s%2B-81+=+0) has the following solutons:

s%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-120%29%5E2-4%2A9%2A-81=17316.

Discriminant d=17316 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--120%2B-sqrt%28+17316+%29%29%2F2%5Ca.

s%5B1%5D+=+%28-%28-120%29%2Bsqrt%28+17316+%29%29%2F2%5C9+=+13.9772373998204
s%5B2%5D+=+%28-%28-120%29-sqrt%28+17316+%29%29%2F2%5C9+=+-0.643904066487102

Quadratic expression 9s%5E2%2B-120s%2B-81 can be factored:
9s%5E2%2B-120s%2B-81+=+9%28s-13.9772373998204%29%2A%28s--0.643904066487102%29
Again, the answer is: 13.9772373998204, -0.643904066487102. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-120%2Ax%2B-81+%29