Question 803108
Let {{{ t }}} = the time in hrs for going upstream
{{{ 8 - t }}} = the time in hrs for going downstream
Let {{{ b }}} = the speed of the boat in still water
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Going upstream:
(1) {{{ 2 = ( b - 3 )*t }}}
Going downstream:
(2) {{{ 2 = ( b + 3 )*( 8 - t ) }}}
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(2) {{{ 2 = 8b + 24 - b*t - 3t  }}}
(2) {{{ 2 = b*( 8 - t ) + 24 - 3t }}}
and
(1) {{{ 2 = b*t - 3t }}}
(1) {{{ b*t = 2 + 3t }}}
(1) {{{ b = ( 2 + 3t ) / t }}}
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By substitution:
(2) {{{ 2 = (( 2 + 3t ) / t )*( 8 - t ) + 24 - 3t }}}
(2) {{{ 3t = (( 2 + 3t ) / t )*( 8 - t ) + 22 }}}
Multiply both sides by {{{ t }}}
(2) {{{ 3t^2 = ( 2 + 3t )*( 8 - t ) + 22t }}}
(2) {{{ 3t^2 = 16 + 24t - 2t - 3t^2 + 22t }}}
(2) {{{ 6t^2 - 44t - 16 = 0 }}}
(2) {{{ 3t^2 - 22t - 8 = 0 }}}
Use quadratic formula
{{{ t = ( -b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 3 }}}
{{{ b = -22 }}}
{{{ c = -8 }}}
{{{ t = ( -(-22) +- sqrt( (-22)^2 - 4*3*(-8) )) / (2*3) }}}
{{{ t = ( 22 +- sqrt( 484 + 96 )) / (2*3) }}}
{{{ t = ( 22 +- sqrt( 580 )) / 6 }}}
{{{ t = ( 22 + 24.083 ) / 6 }}}
{{{ t = 46.083 / 6 }}}
{{{ t = 7.681 }}} hrs
and, since
(1) {{{ b = ( 2 + 3t ) / t }}}
(1) {{{ b = ( 2 + 3*7.681 ) / 7.681 }}}
(1) {{{ b = 25.042 / 7.681 }}}
(1) {{{ b = 3.26 }}}
The speed of the boat in still water is 3.26 km/hr
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check:
(2) {{{ 2 = ( b + 3 )*( 8 - t ) }}}
(2) {{{ 2 = ( 3.26 + 3 )*( 8 - 7.681 ) }}}
(2) {{{ 2 = 6.26 * .319 }}}
(2) {{{ 2 = 1.997 }}}
error due to rounding off
OK