document.write( "Question 1155359: At a certain vineyard it is found that each grape vine produces about 10 lb of grapes in a season when about 600 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by
\n" ); document.write( "A(n) = (600 + n)(10 − 0.01n)
\n" ); document.write( "where n is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #777939 by ikleyn(52781) $\"\"$ $\"About$

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

\r\n" );
document.write( "\r\n" );
document.write( "Write the function A(n) as a quadratic function in the standard form\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    A(n) = (600+n)*(10-0.01n) = 6000 + 10n - 6n - 0.01n^2 = - 0.01n^2 + 4n + 6000.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Any quadratic function y(x) = ax^2 + bx + c  with the negative leading coefficient \"a\"  has the maximum at  x = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In this case, the quadratic function A(n) has coefficients  a = -0.01, b = 4.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, it gets the maximum value at n =  =  = 200.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  200 ADDITIONAL vines per acre will provide the maximum of grape production.\r\n" );
document.write( "\r\n" );
document.write( "         In all, 600+200 = 800 vines should be planted.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\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( "    - A rectangle with a given perimeter which has the maximal area is a square\r
\n" ); document.write( "\n" ); document.write( "    - A farmer planning to fence a rectangular garden to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - A farmer planning to fence a rectangular area along the river to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "    - Using quadratic functions to solve problems on maximizing revenue/profit (*)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Do not miss the lesson marked (*) in the list, since it contains many similar solved problems, closed to yours in your post.\r
\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( "Save the link to this online textbook together with its description\r
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );