Question 1058316
Rate of heating for large unit alone:
[ 1 piece of iron melted ] / [ t hrs ]
Rate of heating for small unit alone:
[ 1 piece of iron melted ] / [ t + 56 hrs ]
Rate for both units together:
[ 1 piece of iron melted ] / [ 21 hrs ]
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Add the different units rates of melting
to get rate working together:
{{{ 1/t + 1/( t+56 ) = 1/21 }}}
Multiply both sides by {{{ t*( t+56 )*21 }}}
{{{ 21*( t+56 ) + 21t = t*( t+56 ) }}}
{{{ 21t + 1176 + 21t = t^2 + 56t }}}
{{{ t^2 + 14t - 1176 = 0 }}}
I'll complete square:
{{{ t^2 + 14t + (14/2)^2 = 1176 + (14/2)^2 }}}
{{{ t^2 + 14t + 49 = 1176 + 49 }}}
{{{ ( t + 7 )^2 = 1225 }}}
{{{ ( t + 7 )^2 = 35^2 }}}
{{{ t + 7 = 35 }}}
{{{ t = 28 }}}
and
{{{ t + 56 = 84 }}}
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The smaller unit alone takes 84 hrs
The larger unit alone takes 28 hrs
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check:
{{{ 1/t + 1/( t+56 ) = 1/21 }}}
{{{ 1/28 + 1/84 = 1/21 }}}
{{{ .035714 + .011905 = .047619 }}}
{{{ .047619 = .047619 }}}
OK