Question 1154641
Let {{{ r }}} = rate with tank empty
{{{ r - 10 }}} rate with tank full
Let {{{ t }}} = time in hrs with tank empty
{{{ 3.5 - t }}} = time with tank full
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With tank empty:
(1) {{{ 60 = r*t }}}
With tank full:
(2) {{{ 60 = ( r - 10 )*( 3.5 - t ) }}}
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(1) {{{ t = 60/r }}}
(2) {{{ 60 = ( r - 10 )*( 3.5 - 60/r ) }}}
(2) {{{ 60 = 3.5r - 35 - 60 + 600/r }}}
(2) {{{ 155 = 3.5r + 600/r }}}
(2) {{{ 155r = 3.5r^2 + 600 }}}
(2) {{{ 3.5r^3 - 155r + 600 = 0 }}}
(2) {{{ 35r^2 - 1550r + 6000 = 0 }}}
(2) {{{ 7r^2 - 310r + 1200 = 0 }}}
{{{ r = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{ a = 7 }}}
{{{ b = -310 }}}
{{{ c = 1200 }}}
{{{ r = (-(-310) +- sqrt( (-310)^2-4*7*1200 ))/(2*7) }}}
{{{ r = (310 +- sqrt( 96100 - 33600 ))/14 }}}
{{{ r = (310 +- sqrt( 62500 ))/14 }}}
{{{ r = ( 310 - 250 ) / 14 }}}
{{{ r = 60/14 }}} too slow
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{{{ r = ( 310 + 250 ) / 14 }}}
{{{ r = 560/14 }}}
{{{ r = 40 }}}
The train's speed with tank empty was 40 mi/hr
The train's speed with tank full was 30 mi/hr
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check:
(1) {{{ 60 = r*t }}}
(1) {{{ 60 = 40*t }}}
(1) {{{ t = 1.5 }}}
and
(2) {{{ 60 = ( r - 10 )*( 3.5 - t ) }}}
(2) {{{ 60 = ( 40 - 10 )*( 3.5 - 1.5 ) }}}
(2) {{{ 60 = 30*2 }}}
(2) {{{ 60 = 60 }}}
OK