Question 1050063
Let {{{ t }}} = time spent going downstream
{{{ 1 - t }}} = time spent going upstream
Let {{{ s }}} = the speed of the boat in still water
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Equation for going downstream:
(1) {{{ 1.5 = ( s + 4 )*t }}}
Equation for going upstream:
(2) {{{ 1.5 = ( s - 4 )*( 1 - t ) }}}
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(2) {{{ 1.5 = s - 4 - s*t + 4t }}}
(2) {{{ 1.5 = s - 4 + t*( 4 - s ) }}}
and
(1) {{{ t = 1.5 / ( s + 4 ) }}}
Substitute (1) into (2)
(2) {{{ 1.5 = s - 4 + ( 1.5 / ( s+4 ) )*( 4 - s ) }}}
(2) {{{ 5.5 = s + ( 1.5 / ( s+4 ) )*( 4 - s ) }}}
Multiply both sides by {{{ s+4 }}}
(2) {{{ 5.5*( s+4 ) = s*( s+4 ) + 1.5*( 4 - s ) }}}
(2) {{{ 5.5s + 22 = s^2 + 4s + 6 - 1.5s }}}
(2) {{{ s^2 + 2.5s + 6 = 5.5s + 22 }}}
(2) {{{ s^2 - 3s = 16 }}}
I'll try completing the square:
(2) {{{ s^2 - 3s + ( -3/2)^2 = 16 + (-3/2)^2 }}}
(2) {{{ s^2 - 3s + 9/4 = 16 + 9/4 }}}
(2) {{{ s^2 - 3s + 9/4 = 64/4 + 9/4 }}}
(2) {{{ s^2 - 3s + 9/4 = 73/4 }}}
(2) {{{ ( s - 3/2 )^2 = 73/4 }}}
(2) {{{ s - 3/2 = (1/2)*sqrt(73) }}}
(2) {{{ s = (1/2)*8.544 + 3/2 }}}
(2) {{{ s = 4.272 + 1.5 }}}
(2) {{{ s = 5.772 }}} mi/hr
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check answer:
(1) {{{ t = 1.5 / ( s + 4 ) }}}
(1) {{{ t = 1.5 / ( 5.772 + 4 ) }}}
(1) {{{ t = 1.5/9.772 }}}
(1) {{{ t = .1535 }}} hrs
and
(2) {{{ 1.5 = ( 5.772 - 4 )*( 1 - t ) }}}
(2) {{{ 1.5 = 1.772*( 1 - t ) }}}
(2) {{{ 1.5 = 1.772 - 1.772t }}}
(2) {{{ 1.772t = .272 }}}
(2) {{{ t = .1535 }}} hrs
It's about a 9 minute race! ( if I'm right )