Question 545887
Let {{{w}}} = rate of current
Let {{{b}}} = rate of boat in still water
{{{ b - w }}} = rate going against current
{{{ b + w }}} = rate going with current
Let {{{t}}} = time for boat to go {{{59}}} mi with current
-------------
given:
equation for going downstream ( with current )
{{{ 59 = ( b + w )* t }}}
{{{ 59 = ( 14 + w )*t }}}
(1) {{{ t = 59 / ( 14 + w ) }}}
equation for going upstream ( against current )
{{{ 59 = ( b - w )*( t + 6 ) }}}
{{{ 59 = ( 14 - w )*(t + 6 ) }}}
{{{ 59 = 14t - w*t + 84 - 6w }}}
(2) {{{ t*( w - 14 ) + 6w = 84 - 59 }}}
(2) {{{ t*( w - 14 ) + 6w = 25 }}}
------------------
Substitute (1) into (2)
(2) {{{ ( 59/( 14 + w ))*( w - 14 ) + 6w = 25 }}}
multiply both sides by {{{ 14 + w ) }}}
(2) {{{ 59*( w - 14 ) + 6c*( 14 + w ) = 25*( 14 + w ) }}}
(2) {{{ 59w - 826 + 84w + 6w^2 = 350 + 25w }}}
(2) {{{ 6w^2 + 84w + 59w - 25w - 825 - 350 = 0 }}}
(2) {{{ 6w^2 + 118w - 1175 = 0 }}}
Use quadratic equation
{{{w = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{ a = 6 }}} 
{{{ b = 118 }}}
{{{ c = -1175 }}}
{{{w = (-118  +- sqrt( 118^2-4*6*(-1175) ))/(2*6) }}}
{{{w = (-118  +- sqrt( 13924 + 28200 )) / 12 }}}
{{{w = (-118  +- sqrt( 42124 )) / 12 }}}
{{{ w = ( -118 + 205.24) / 12 }}}
{{{ w = 87.24/12 }}}
{{{ w = 7.27 }}}
The rate of the current is 7.27 mi/hr
check answer:
{{{ 59 = ( 14 + w )*t }}}
{{{ 59 = ( 14 + 7.27 )*t }}}
{{{ 59 = 21.27t }}}
{{{ t = 2.774 }}}
and
{{{ 59 = ( 14 - w )*(t + 6 ) }}}
{{{ 59 = ( 14 - 7.27 )*(t + 6 ) }}}
{{{ 59 = 6.73*( t + 6 ) }}}
{{{ 59 = 6.73t + 40.38 }}}
{{{ 6.73t = 59 - 40.38 }}}
{{{ 6.73t = 18.62 }}}
{{{ t = 2.767 }}}
This looks close enough