Question 728566
Since you're only dealing with 1000 bottles, you can
say that 1000 bottles = 1 job
Add the rates that the 2 machines work at
( 1 job / 10 min ) + ( 1 job / 8 min ) = 1 job / t min
{{{ 1/10 + 1/8 = 1/t }}}
Multiply both sides by {{{ 40t }}}
{{{ 4t + 5t = 40 }}}
{{{ 9t = 40 }}}
{{{ t = 4.444 }}} min with both working
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{{{ t[1] / t[2] = 10/8 }}}
{{{ t[2] = (8/10)*t[1] }}}
{{{ t[2] = (4/5)*t[1] }}}
{{{ 1/t[1] + 1/t[2] = 1/4.444 }}}
{{{ 1/t[1] + 1/t[2] = 1/4.444 }}}
{{{ 1/t[1]  + 1/((4/5)*t[1]) = 9/40 }}}
{{{ ( 1/t[1] )*( 1 + 5/4 ) = 9/40 }}}
{{{ t[1] = (40/9)*(9/4) }}}
{{{ t[1] = 10 }}} min
{{{ t[2] = (4/5)*t[1] }}}
{{{ t[2] = (8/10)*10 }}}
{{{ t[2] = 8 }}} min
I'm not sure this is what they want, though
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{{{ 1/(a-2) + 1/(a^2 - 2a ) = 9/40 }}}
Multiply both sides by {{{ 40*(a-2)*(a^2 - 2a) }}}
{{{ 40*( a^2 - 2a ) + 40*( a-2 ) = 9*( a-2 )*( a^2 - 2a ) }}}
{{{ 40*( a^2 -a - 2 ) = 9*( a^3 - 2a^2 - 2a^2 - 4a ) }}}
{{{ 40a^2 - 40a - 80 = 9a^3 - 36a^2 - 36a }}}
{{{ 9a^3 - 76a^2 + 4a + 80 = 0 }}}
Hope I got it
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Tough problems