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The publisher of a magazine that has a circulation of 80,000 and sells for 1.60$ a copy decides to raise the price of the magazine because of costs. By surveying the readers, he finds that he will lose 10,000 readers for each .40 $ increase. What price per copy maximizes growth?
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\n" ); document.write( "By growth do they mean maximize revenue, if that is the case we can do it this way;
\n" ); document.write( ":
\n" ); document.write( "Let x = no. of 40 cent increases
\n" ); document.write( "and
\n" ); document.write( "Let x = no. of 10,000 reader decreases
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\n" ); document.write( "Revenue = price * no. of readers
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\n" ); document.write( "R = (1.60 + .40x) * (80000 - 10000x)
\n" ); document.write( "FOIL
\n" ); document.write( "R = 128000 - 16000x + 32000x - 4000x^2
\n" ); document.write( "As a quadratic
\n" ); document.write( "-4000x^2 + 16000x + 128000 = 0
\n" ); document.write( ":
\n" ); document.write( "Simplify, divide equation by 4000
\n" ); document.write( "-x^2 + 4x + 32 = 0
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\n" ); document.write( "Max will occur at the axis of symmetry: x = -b/(2a); Here: a=-1; b=4
\n" ); document.write( "x =
\n" ); document.write( "x = +2
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\n" ); document.write( "That would be 2 ea .40 increases for a cost of 2.40 per copy
\n" ); document.write( "and
\n" ); document.write( "2 ea 10000 readers lost for a total 60000 readers
\n" ); document.write( ":
\n" ); document.write( "Revenue = 60000 * 2.40 = $144,000 (maximum)
\n" ); document.write( ":
\n" ); document.write( "If you graphed it:
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\n" ); document.write( "Multiply the y value by 4000 to get the actual revenue \n" ); document.write( "