SOLUTION: 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 plante

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
A(n) = (600 + n)(10 − 0.01n)
where n is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

Answer by ikleyn(52776) (Show Source): You can put this solution on YOUR website!
.


Write the function A(n) as a quadratic function in the standard form


    A(n) = (600+n)*(10-0.01n) = 6000 + 10n - 6n - 0.01n^2 = - 0.01n^2 + 4n + 6000.


Any quadratic function y(x) = ax^2 + bx + c  with the negative leading coefficient "a"  has the maximum at  x = .


In this case, the quadratic function A(n) has coefficients  a = -0.01, b = 4.


Therefore, it gets the maximum value at n =  =  = 200.


ANSWER.  200 ADDITIONAL vines per acre will provide the maximum of grape production.

         In all, 600+200 = 800 vines should be planted.

Solved.

-------------------

On finding the maximum/minimum of a quadratic function see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola

    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area

    - Using quadratic functions to solve problems on maximizing revenue/profit (*)

Do not miss the lesson marked (*) in the list, since it contains many similar solved problems, closed to yours in your post.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.

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