Question 1010003
Let {{{ s }}} = the speed she can paddle in still water in km/hr
{{{ s + 2 }}} = the speed of the kayak going downstream in km/hr
{{{ s - 2 }}} = the speed of the kayak going upstream in km/hr
Let {{{ t }}} = her time in hrs to travel 15 km upstream
{{{ 4 - t }}} = her time in hrs to travel 15 km downstream
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Her equation for paddling upstream:
(1) {{{ 15 = ( s - 2 )*t }}}
Her equation for paddling downstream:
(2) {{{ 15 = ( s + 2 )*( 4 - t ) }}}
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(2) {{{ 15 = 4s + 8 - ( s + 2 )*t }}}
and
(1) {{{ t = 15/( s - 2 ) }}}
Substitute (1) into (2)
(2) {{{ 15 = 4s + 8 - ( s + 2 )*( 15/( s - 2 ) ) }}}  
(2) {{{ 7 = 4s -  ( s + 2 )*( 15/( s - 2 ) ) }}}  
Multiply both sides by {{{ s - 2 }}}
(2) {{{ 7*( s - 2 ) = 4s*( s - 2 ) - 15*( s + 2 ) }}}
(2) {{{ 7s - 14 = 4s^2 - 8s - 15s - 30 }}}
(2) {{{ 4s^2 - 30s - 16 = 0 }}}
(2) {{{ 2s^2 - 15s - 8 = 0 }}}
Use the quadratic formula
{{{ s = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}} 
{{{ a = 2 }}}
{{{ b = -15 }}}
{{{ c = -8 }}}
{{{ s = (-(-15) +- sqrt( (-15)^2 - 4*2*(-8) )) / (2*2) }}} 
{{{ s = ( 15 +- sqrt( 225 + 64 )) / 4 }}} 
{{{ s = ( 15 +- sqrt( 289 )) / 4 }}} 
{{{ s = ( 15 + 17 ) / 4 }}}
{{{ s = 32/4 }}}
{{{ s = 8 }}}
The speed she can paddle in still water is 8 km/hr
check:
(1) {{{ 15 = ( 8 - 2 )*t }}}
(1) {{{ 15 = 6t }}}
(1) {{{ t = 2.5 }}} hrs
and
(2) {{{ 15 = ( 8 + 2 )*( 4 - t ) }}}
(2) {{{ 15 = 10*( 4 - t ) }}}
(2) {{{ 15 = 40 - 10t }}}
(2) {{{ 10t = 25 }}}
(2) {{{ t = 2.5 }}} hrs
OK