SOLUTION: During the first part of a trip, a canoeist travels 83 miles at a certain speed. The canoeist travels 6 miles on the second part of the trip at a speed 5 mph slower. The total time

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Question 471037: During the first part of a trip, a canoeist travels 83 miles at a certain speed. The canoeist travels 6 miles on the second part of the trip at a speed 5 mph slower. The total time for the trip is 2 hours. What was the on each part of the trip?
The speed of the first part of the trip was ___mph.
(Simplify your answer. Type an integer or a decimal. Round to the nearest hundredth.)
The speed of the second part of the trip was __ mph.
(Simplify your answer. Type an integer or a decimal. Round to the nearest hundredth.)
Found 2 solutions by jorel1380, ankor@dixie-net.com:
Answer by jorel1380(3719) About Me  (Show Source):
You can put this solution on YOUR website!
83/(n+5)+6/n=2
83n+6n+30=2n2+10n
2n2-79n+30=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-79x%2B30+=+0) has the following solutons:

x%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=%28-79%29%5E2-4%2A2%2A30=6001.

Discriminant d=6001 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--79%2B-sqrt%28+6001+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-79%29%2Bsqrt%28+6001+%29%29%2F2%5C2+=+39.1165304068643
x%5B2%5D+=+%28-%28-79%29-sqrt%28+6001+%29%29%2F2%5C2+=+0.383469593135686

Quadratic expression 2x%5E2%2B-79x%2B30 can be factored:
2x%5E2%2B-79x%2B30+=+2%28x-39.1165304068643%29%2A%28x-0.383469593135686%29
Again, the answer is: 39.1165304068643, 0.383469593135686. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-79%2Ax%2B30+%29

The answer is 39.1165304068643 mph or 0.383469593135686 mph. Throwing out the second answer as impractical, we get the speed going as 44.1165304068643 mph, and the speed going back as 39.1165304068643 mph..

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
During the first part of a trip, a canoeist travels 83 miles at a certain speed.
The canoeist travels 6 miles on the second part of the trip at a speed 5 mph slower.
The total time for the trip is 2 hours.
What was the on each part of the trip?
:
Let s = speed on the 1st part of the trip
It says,"the second part of the trip at a speed 5 mph slower.", therefore
(s-5) = speed on the 2nd part
:
Write a time equation, time = dist/speed
:
1st part time + 2nd part time = 2 hrs
83%2Fs + 6%2F%28%28s-5%29%29 = 2
:
multiply by s(s-5), results:
83(s-5) + 6s = 2s(s-5)
:
83s - 415 + 6s = 2s^2 - 10s
:
89s - 415 = 2s^2 - 10s
Arrange as a quadratic equation
0 = 2s^2 - 10s - 89s + 415
:
2s^2 - 99s + 415 = 0
Use the quadratic formula to find s
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
In this equation: x=s; a=2; b=-99; c=415
s+=+%28-%28-99%29+%2B-+sqrt%28-99%5E2-4%2A2%2A415+%29%29%2F%282%2A2%29+
:
s+=+%2899+%2B-+sqrt%289801-3320+%29%29%2F4+
:
s+=+%2899+%2B-+sqrt%286481%29%29%2F4+
Two solutions
s+=+%2899+-+80.5%29%2F4+
s = 18.5%2F4
s = 4.625, obviously this is not the solution
and
s+=+%2899+%2B+80.5%29%2F4+
s = 179.5%2F4
s = 44.87 mph, is the speed on the 1st part of the trip
and
44.88 - 5 = 39.87 mph is the speed on the 2nd part
:
:
:
Check this by finding the actual time of each part
83/44.87 = 1.85 hrs
6/39.87 = 0.15 hrs
----------------------
total time: 2 hrs