document.write( "Question 964813: Two numbers are related by the equation 3x+ 2y = 42. Find the values of the numbers so that their product is a minmum \n" ); document.write( "

Algebra.Com's Answer #589720 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
Find the minimum value of Z where Z is defined as,
\n" ); document.write( " $\"Z=xy\"$
\n" ); document.write( ".
\n" ); document.write( ".
\n" ); document.write( "From the equation,
\n" ); document.write( " $\"2y=42-3x\"$
\n" ); document.write( " $\"y=21-%283%2F2%29x\"$
\n" ); document.write( "Substituting,
\n" ); document.write( " $\"Z=x%2821-%283%2F2%29x%29\"$
\n" ); document.write( " $\"Z=21x-%283%2F2%29x%5E2\"$
\n" ); document.write( "Convert Z to vertex form, then you can find the extremum.
\n" ); document.write( " $\"Z=-%283%2F2%29x%5E2%2B21x\"$
\n" ); document.write( " $\"Z=-%283%2F2%29%28x%5E2-14x%29\"$
\n" ); document.write( " $\"Z=-%283%2F2%29%28x%5E2-14x%2B49%29%2B%283%2F2%29%2849%29\"$
\n" ); document.write( " $\"Z=-%283%2F2%29%28x-7%29%5E2%2B147%2F2\"$
\n" ); document.write( "The value at the vertex is a maximum because of the $\"-3%2F2\"$ multiplier.
\n" ); document.write( "It occurs when $\"x=7\"$ and the maximum value is $\"147%2F2\"$ .
\n" ); document.write( "There is no minimum for the product, only a maximum.
\n" ); document.write( "Check your problem setup and repost if needed.
\n" ); document.write( "

.\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );