document.write( "Question 1183271: Gemma's mother owns a manufacturing company that produces key rings. Last year, she collected data about the
\n" ); document.write( "number of key rings produced per day and the corresponding profit. The data can be modeled by the relation
\n" ); document.write( "𝑷 = −𝟐𝒌^𝟐 + 𝟏𝟐𝒌 − 𝟏𝟎, where 𝑷 is the profit in thousands of dollars and 𝒌 is the number of key rings in
\n" ); document.write( "thousands. You must solve the following questions by factoring.
\n" ); document.write( "a) How many key rings must be produced so that there is no profit and no loss [𝑷 = 𝟎]?
\n" ); document.write( "b) How many key rings must be produced for the maximum profit?
\n" ); document.write( "c) What is the maximum profit? \n" ); document.write( "

Algebra.Com's Answer #813497 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Gemma's mother owns a manufacturing company that produces key rings. Last year, she collected data about the
\n" ); document.write( "number of key rings produced per day and the corresponding profit. The data can be modeled by the relation
\n" ); document.write( "𝑷 = −𝟐𝒌^𝟐 + 𝟏𝟐𝒌 − 𝟏𝟎, where 𝑷 is the profit in thousands of dollars and 𝒌 is the number of key rings in
\n" ); document.write( "thousands. You must solve the following questions by factoring.
\n" ); document.write( "a) How many key rings must be produced so that there is no profit and no loss [𝑷 = 𝟎]?
\n" ); document.write( "b) How many key rings must be produced for the maximum profit?
\n" ); document.write( "c) What is the maximum profit?
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document.write( "(a)  To answer, you should solve the equation  P = 0, which is\r\n" );
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document.write( "         -2k^2 + 12k - 10 = 0,\r\n" );
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document.write( "     or, EQUIVALENTLY,  \r\n" );
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document.write( "         k^2 - 6k + 5 = 0.\r\n" );
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document.write( "     Factor\r\n" );
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document.write( "         (k-5)*(k-1) = 0\r\n" );
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document.write( "     and find two roots,  5 and 1.\r\n" );
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document.write( "     This question has two answers:  1 thousand keys and 5 thousand keys.\r\n" );
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document.write( "(b)  and  (c)   The maximum profit corresponds to the vertex of the quadratic function.\r\n" );
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document.write( "         -2k^2 + 12k - 10 = -2*(k^2 - 6k + 5) = -2*((k-3)^2 - 4).\r\n" );
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document.write( "      The vertex is at  k= 3,  with the value of the quadratic function equal to  (-2)*(-4) = 8.\r\n" );
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document.write( "      THEREFORE, the answer to question (b) is 3000 keys (the optimal production) and  8000 dollars (the maximum profit).\r\n" );
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