At a certain vineyard it is found that each grape vine produces about 10 lb of grapes in a season when about 700 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) = (700 + 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:Number of vines that should be planted are 150.Step-by-step explanation:The number of pounds of pounds of grapes produced per acre is represented by the expression [tex]A_{n}=(700+n)(10-0.01n)[/tex]Where n = additional vines plantedTo maximize the production of grapes we will find the derivative of A(n) and equate it to zero.[tex]A_{n}=(7000+3n-0.01n^{2} )[/tex][tex]A'_{n}=(3-0.02n)[/tex]For [tex]A'_{n}=0[/tex]3 - 0.02n = 00.02n = 3n = [tex]\frac{3}{0.02}[/tex]n = 150 To check whether the maximum value of the function is at n = 150, we will find the second derivative A(n).[tex]A''_{n}=-0.02[/tex]Which shows A"(n) < 0Therefore, A(n) has the maximum value at n = 150.Therefore, number of vines that should be planted are 150.