document.write( "Question 1118948: A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $5, and the society sells an average of 30 feeders per week at a price of $9 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 3 sales per week.
\n" ); document.write( "(a) Find a function P that models weekly profit in terms of price per feeder. (Let x be the price per feeder.)
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Algebra.Com's Answer #734373 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities.
\n" ); document.write( "The materials for each feeder cost $5, and the society sells an average of 30 feeders per week at a price of $9 each.
\n" ); document.write( "The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 3 sales per week.\r
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\n" ); document.write( "\n" ); document.write( "(a) Find a function P that models weekly profit in terms of price per feeder. (Let x be the price per feeder.)\r
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\n" ); document.write( "\n" ); document.write( " * * * As the problem is worded, it is much easier to me to use x for the exceed the price over 9 dollars. * * * \r
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document.write( "         >>> I don't like VERY MUCH when somebody comes for help and starts teach me how to solve the problem.\r\n" );
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document.write( "         >>> Especially when your advice makes the solution harder . . . \r\n" );
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document.write( "         >>> Let me decide how to do it better - or solve the problem on your own . . . \r\n" );
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document.write( "     The phrase \" . . . for every dollar increase, it loses 3 sales per week.\" means that at the price (9+x) dollars the number of sales is (30-3x) feeders per week.\r\n" );
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document.write( "     In  turn, it means that the profit function per week as the function of x  is \r\n" );
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document.write( "        P(x) = (9+x)*(30-3x) - 5*(30-3x)   (1)   (profit = revenue minus cost)\r\n" );
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document.write( "     It is the answer to question (a).\r\n" );
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document.write( "     \r\n" );
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document.write( "(b)  What price should the society charge for each feeder to maximize profits?\r\n" );
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document.write( "      All you need is to simplify (1) and find the maximum of this quadratic function:\r\n" );
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document.write( "      P(x) = 270 + 30x - 27x - 3x^2 - 150 + 15x = -3x^2 + 18x + 120.\r\n" );
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document.write( "      For a quadratic function q(x) = ax^2 + bx + c   with negative leading coefficient \"a\" the maximum is achieved at x = .  \r\n" );
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document.write( "      In your case  a = -3,  b = 18,  so  x =  = 3.\r\n" );
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document.write( "      So, the optimal price is  9+3 = 12 dollars  per feeder.\r\n" );
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document.write( "(c)  What is the maximum weekly profit?\r\n" );
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document.write( "     The current weekly profit is  30*$9 - 30*5 = 120 dollars.\r\n" );
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document.write( "     The projected (predicted) weekly profit is  (see formula (1) )\r\n" );
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document.write( "        (9+3)*(30-3*3) - 5(30-3*3) = 147 dollars.\r\n" );
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\n" ); document.write( "\n" ); document.write( "---------------\r
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\n" ); document.write( "\n" ); document.write( "If you want to see other similar solved problems, look into the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Using quadratic functions to solve problems on maximizing revenue/profit\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "My other lessons in this site on finding the maximum/minimum of a quadratic function are \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
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\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
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\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
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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