Question 780896
At exactly 11 o'clock. the hands are
5 minutes apart. That is {{{ 5/60 = 1/12 }}}
of a full circle, and that is {{{ 30 }}} degrees
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The minute hand rotates at a rate of {{{ 360/60 }}}
degrees / minute. or
{{{ 6 }}} degrees / minute
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The hour hand rotates at a rate of 
{{{ ((5/60)*360) / 60 }}} degrees/minute
{{{ 30/60 = 1/2 }}} degrees / minute
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I want to find when the differences in the positions
of the hands is {{{ 270 }}} degrees, which is the
2nd time that they are {{{ 90 }}} degrees apart
( {{{ 360 - 270 = 90 }}} )
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Let {{{ t }}} = time in minutes
For the minute hand:
position = ( degrees/min x minutes ) + initial difference
position = {{{ 6t + 30 }}}
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For the hour hand:
position = {{{ (1/2)*t }}}
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Let the difference = {{{ 270 }}} degees
{{{ ( 6t + 30 ) - (1/2)*t = 270 }}}
{{{ 6t + 30 - (1/2)*t = 270 }}}
{{{ 5.5t = 240 }}}
{{{ t = 240 / 5.5 }}}
{{{ t = 43.636 }}}
{{{ .636*60 = 38.2 }}}
The hands will be 90 degrees apart for
the 2nd time in 43 min 38.2 sec
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Hope I got it- looks about right