Question 1060571
These problems are in the category of
" related rate " problems. For word problems,
it's very important to identify their category.
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(1)
What is the head start in miles for the ship?
{{{ d[1] = 28*6.5 }}}
{{{ d[1] = 182 }}} mi
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Let {{{ d }}} = the distance in miles the plane 
travels until it catches up with the ship
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Let {{{ s }}} = the flying speed of the plane so 
that the plane catches up with the ship in
{{{ 1.25 }}} hrs
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Equation for the plane:
(1) {{{ d = s*1.25 }}}
Equation for the ship:
(2) {{{ d - 182 = 28*1.25 }}}
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Substitute (1) into (2)
(2) {{{ 1.25s - 182 = 28*1.25 }}}
(2) {{{ 1.25s = 35 + 182 }}}
(2) {{{ 1.25s = 217 }}}
(2) {{{ s = 173.6 }}}
The plane's flying rate is 173.6 mi/hr
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check the answer:
(1) {{{ d = s*1.25 }}}
(1) {{{ d = 173.6*1.25 }}}
(1) {{{ d = 217 }}}
and
(2) {{{ d - 182 = 28*1.25 }}}
(2) {{{ 217 - 182 = 28*1.25 }}}
(2) {{{ 35 = 35 }}}
OK
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(2)
Let {{{ t }}} = his flying time in hrs
{{{ 4 - t }}} = his time traveling by train
Let {{{ d }}} = the distance to the city
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Equation for traveling by train:
(1) {{{ d = 60*( 4 - t ) }}}
Equation for flying:
(2) {{{ d = 260t }}}
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substitute (2) into (1)
(1) {{{ 260t = 60*( 4 - t ) }}}
(1) {{{ 260t = 240 - 60t }}}
(1) {{{ 320t = 240 }}}
(1) {{{ t = .75 }}} hrs
His flying time is 3/4 of an hour, or 45 minutes
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check the answer
(2) {{{ d = 260t }}}
(2) {{{ d = 260*.75 }}}
(2) {{{ d = 195 }}} mi
and
(1) {{{ d = 60*( 4 - t ) }}}
(1) {{{ d = 60*( 4 - .75 ) }}}
(1) {{{ d = 60*3.25 }}}
(1) {{{ d = 195 }}} mi
OK
Hope this helps