Question 159082
For the two parts of the trip:
{{{d[1] = r[1]*t[1]}}}
{{{d[2] = r[2]*t[2]}}}
given:
{{{d[1] = 75}}}mi
{{{d[2] = 18}}}mi
{{{r[2] = r[1] - 5}}}
{{{t[1] + t[2] = 3}}}hr
{{{t[2] = 3 - t[1]}}}hr
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{{{d[1] = r[1]*t[1]}}}
{{{75 = r[1]*t[1]}}}
(1){{{t[1] = 75 / r[1]}}}
and
{{{d[2] = r[2]*t[2]}}}
{{{18 = (r[1] - 5)*(3 - t[1])}}}
{{{18 = 3r[1] - 15 - r[1]*t[1] + 5t(1)}}}
Note that {{{r[1]*t[1] = 75}}} (given)
{{{18 = 3r[1] - 15 - 75 + 5t[1]}}}
{{{3r[1] + 5t[1] = 108}}}
from (1)
{{{3r[1] + 5*(75/r[1]) = 108}}}
multiply both sides by {{{r[1]}}}
{{{3(r[1])^2 + 375 = 108r[1]}}}
subtract {{{108r[1]}}} from both sides
divide both sides by {{{3}}}
{{{(r[1])^2 - 36r[1] + 125 = 0}}}
use the quadratic formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
where
{{{a = 1}}}
{{{b = -36}}}
{{{c = 125}}}
{{{r[1] = (-(-36) +- sqrt( (-36)^2-4*1*125 ))/(2*1) }}}
{{{r[1] = (36 +- sqrt( 1296 - 500 )) / 2 }}}
{{{r[1] = (36 +- sqrt(796)) / 2 }}}
{{{r[1] = (36 + 28.21) / 2 }}}
{{{r[1] = (36 - 28.21) / 2 }}}
{{{r[1] = 64.21/2}}}
{{{r[1] = 32.1}}}mi/hr
Using the (-) sign, the other answer is
{{{r[1] = 3.90}}}mi/hr (makes no sense, too small)
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check the 1st answer:
{{{r[2] = r[1] - 5}}}
{{{r[2] = 27.1}}}mi/hr
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{{{75 = 32.1*t[1]}}}
{{{t[1] = 2.34}}}hr
{{{18 = r[2]*t[2]}}}
{{{18 = 27.1*t[2]}}}
{{{t[2] = .664}}}hr
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The total time for the trip must be {{{3}}}hr
{{{t[1] + t[2] = 3}}}
{{{2.34 + .664 = 3}}}
{{{3.004 = 3}}} close enough
The speed for the 1st part was 32.1 mi/hr
The speed for the 2nd part was 27.1 mi/hr