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
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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.
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.
