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?
:
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/s}}} + {{{6/((s-5))}}} = 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 = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
In this equation: x=s; a=2; b=-99; c=415
{{{s = (-(-99) +- sqrt(-99^2-4*2*415 ))/(2*2) }}}
:
{{{s = (99 +- sqrt(9801-3320 ))/4 }}}
:
{{{s = (99 +- sqrt(6481))/4 }}}
Two solutions
{{{s = (99 - 80.5)/4 }}}
s = {{{18.5/4}}}
s = 4.625, obviously this is not the solution
and
{{{s = (99 + 80.5)/4 }}}
s = {{{179.5/4}}}
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
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total time: 2 hrs