Question

At a certain vineyard it is found that each grape vine produces about 10 pounds 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 1

Answer:

The number of additional vines to plant in order to maximize production  is 200

The total number of vines to maximize production is 800

Step-by-step explanation:

Notice that the expression for the number of pounds of grapes per acre  is in fact a quadratic expression in the variable "n":

$A(n)=(600+n)(10-0.01n)\\A(n)= 6000-6n+10n-0.01n^2\\A(n) = 6000+4n-0.01n^2$

which is clearly associated with a parabola with arms pointing down (negative coefficient in the variable "n" squared). So by finding the vertex of the parabola, we can give the answer to what "n" (number of additional vines to plant).

Recall that the vertex of a parabola of the general shape: $y=ax^2+bx+c$ has an x-component defined by:

$x_{vertex}=\frac{-b}{2a}$

then in our case, the "n" value for that vertex is: $n_{vertex}= \frac{-4}{2(-0.01)} =200$

Then the additional number of vines to maximize grape production is 200

So the total should be: 600 + 200 =800