document.write( "Question 954660: Find two numbers such that their product is as large as possible, given that two times the first number plus
\n" ); document.write( "three times the second number is 60 \n" ); document.write( "

Algebra.Com's Answer #583098 by Fombitz(32388) $\"\"$ $\"About$

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Let the two numbers be X and Y.
\n" ); document.write( "You want to maximize $\"Z=X%2AY\"$ with $\"2X%2B3Y=60\"$ .
\n" ); document.write( "From the second equation,
\n" ); document.write( " $\"2X=60-3Y\"$
\n" ); document.write( " $\"X=30-%283%2F2%29Y\"$
\n" ); document.write( "Substitute into the product,
\n" ); document.write( " $\"Z=%2830-%283%2F2%29Y%29Y\"$
\n" ); document.write( "Now the product is the function of only one variable.
\n" ); document.write( "You can either take the derivative and find the extrema or since it's a quadratic, you can convert it to vertex form to find the maximum.
\n" ); document.write( " $\"Z=-%283%2F2%29Y%5E2%2B30Y\"$
\n" ); document.write( " $\"Z=-%283%2F2%29%28Y%5E2-20Y%29\"$
\n" ); document.write( " $\"Z=-%283%2F2%29%28Y%5E2-20Y%2B100%29%2B%283%2F2%29%28100%29\"$
\n" ); document.write( " $\"Z=-%283%2F2%29%28Y-10%29%5E2%2B150\"$
\n" ); document.write( "In vertex form, the parabola opens downwards and the vertex value is the maximum.
\n" ); document.write( "The maximum $\"Z=150\"$ occurs when $\"Y=10\"$
\n" ); document.write( "So then,
\n" ); document.write( " $\"X=30-%283%2F2%29%2810%29\"$
\n" ); document.write( " $\"X=30-15\"$
\n" ); document.write( " $\"X=15\"$
\n" ); document.write( " \n" ); document.write( "

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