Question 769537
Let {{{ w }}} = the rate of the wind
Flying with the wind, the plane's rate is:
{{{ 225 + w }}}
Flying against the wind, the plane's rate is:
{{{ 225 - w }}}
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Equation flying with the wind:
(1) {{{ 500 = ( 225 + w )*t }}}
Flying against the wind:
(2) {{{ 500 = ( 225 - w )*( t + .5 ) }}}
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(2) {{{ 500 = 225t - w*t + .5*225 - .5w }}}
(2) {{{ w*t + .5w = 225t + 112.5 - 500 }}}
(2) {{{ w*( t + .5 ) = 225t - 387.5 }}}
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(1) {{{ 500 = 225t + w*t }}}
(1) {{{ w*t = 500 - 225t }}}
(1) {{{ w = 500/t - 225 }}}
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Substitute (1) into (2)
(2) {{{ ( 500/t - 225 )*( t + .5 ) = 225t - 387.5 }}}
(2) {{{ 500 - 225t + 250/t - 112.5 = 225t - 387.5 }}}
(2) {{{ 250/t = 450t - 775 }}}
(2) {{{ 250 = 450t^2 - 775t }}}
(2) {{{ 450t^2 - 775t - 250 = 0 }}}
(2) {{{ 18t^2 - 31t - 10 = 0 }}}
Use quadratic formula
{{{ t = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 18 }}}
{{{ b = -31 }}}
{{{ c = -10 }}}
{{{ t = (-(-31) +- sqrt( (-31)^2 - 4*18*(-10) )) / (2*18) }}}
{{{ t = ( 31 +- sqrt( 961 + 720 )) / 36 }}}
{{{ t = ( 31 +- sqrt( 1681 )) / 36 }}}
{{{ t = ( 31 + 41 ) / 36 }}} ( can't use the negative root )
{{{ t = 72/36 }}}
{{{ t = 2 }}}
and, since
(1) {{{ 500 = ( 225 + w )*t }}}
(1) {{{ 500 = ( 225 + w )*2 }}}
(1) {{{ 450 + 2w = 500 }}}
(1) {{{ 2w = 50 }}}
(1) {{{ w = 25 }}}
The rate of the wind is 25 mi/hr
check:
(2) {{{ 500 = ( 225 - w )*( t + .5 ) }}}
(2) {{{ 500 = ( 225 - 25 )*( 2 + .5 ) }}}
(2) {{{ 500 = 200*2.5 }}}
(2) {{{ 500 = 500 }}}
OK