document.write( "Question 1142726: **Solve using calculus optimization**\r
\n" ); document.write( "\n" ); document.write( "A company that produces chemicals used in manufacturing sells a certain product for $200 per unit. The total production cost for the product is\r
\n" ); document.write( "\n" ); document.write( "C(x)= 500,000 + 80x + 0.003x^2\r
\n" ); document.write( "\n" ); document.write( "The company can produce no more than 30,000 units during a given time period. How
\n" ); document.write( "many units of the chemical product should be manufactured and sold to maximize profit? \n" ); document.write( "

Algebra.Com's Answer #763460 by ikleyn(52780) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "The PROFIT function P(x) is the difference  REVENUE - COST \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    P(x) = 200x - (500000+80x+0.003x^2) = -0.003x^2 + 120x - 500000.     (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This quadratic function plot is opened downward, so it has the maximum.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The maximum of a quadratic function is achieved at  x =   (referring to the general form f(x) = ax^2 + bx + c).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In our case  \" a \" = -0.003, b = 120,  so the maximum is achieved at  x =  =  = 20000.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is  LESS  than the company capacity of  30000.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the company profit will be maximum at x = 20000.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The profit value is then  \r\n" );
document.write( "\r\n" );
document.write( "     = 700,000.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  20,000 units should be manufactured and sold.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "-----------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On finding the maximum/minimum of a quadratic function see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the vertex of a parabola\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this textbook under the topic \"Finding minimum/maximum of quadratic functions\". \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "================\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Sorry, in my solution I used the standard Algebra approach instead of Calculus.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "But you will get the same answer, using Calculus.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "For it, simply differentiate the quadratic function P(x) (formula (1)) and equate the derivative to zero \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "    P'(x) = -0.003*2*x + 120 = 0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    120 = 0.006x\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    x =  = 20000.       ANSWER\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved. // So, now you have both Algebra and Calculus solutions.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );