Question 24847
Let x = rate of the elevator going up
x+1 = rate of the elevator going down.


Since the elevator makes the entire trip in 2 minutes (i.e., 120 seconds) with a 24 second stop at the top, it actually spends a total of 96 seconds moving up and down.  


Basic formula: D= RT, from which {{{T = D/R}}}.


Time going up ={{{ D/R = 140/x}}}
Time going down = {{{D/R = 140/(x+1)}}}


The equation is based upon the total time = 96 seconds:
{{{T[up] + T[down] = 96}}}
{{{140/x + 140/(x+1) = 96}}}


The LCD = x(x+1), so multiply both sides of this fractional equation by x(x+1):

{{{x*(x+1)*(140/x) + x*(x+1)*(140/(x+1)) = 96x*(x+1) }}}


All the denominator factors divide out leaving:
{{{140(x+1) + 140x = 96x(x+1) }}}
{{{140x + 140 + 140x = 96x^2 + 96x }}}
{{{280x + 140 = 96x^2 + 96x}}}


This is a quadratic, which you must set equal to zero:
{{{0 = 96x^2 +96x - 280x - 140 }}}
{{{0= 96x^2 - 184x - 140 }}}


Take out a common factor of 4:
{{{0 = 4(24x^2 - 46x -35 )}}}


Either solve by factoring or quadratic formula or graphing calculator methods, take your pick.  IT DOES FACTOR, believe it or not!!

{{{0 = 4(12x + 7 )(2x -5) }}}

{{{x = -7/12}}} Reject the negative answer!
{{{x = 5/2 }}} or 2.5 m/sec UP
{{{x+1= 3.5}}} m/sec DOWN.


NICE PROBLEM!!


R^2 at SCC