Question

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 1

Number of vines that should be planted are 150.

Explanation:

The number of pounds of pounds of grapes produced per acre is represented by the expression
`A_(n)=(700+n)(10-0.01n)`

Where n = additional vines planted

To maximize the production of grapes we will find the derivative of A(n) and equate it to zero.

`A_(n)=(7000+3n-0.01n^(2) )`

`A'_(n)=(3-0.02n)`

For
`A'_(n)=0`

3 - 0.02n = 0

0.02n = 3

n =
`(3)/(0.02)`

n = 150

To check whether the maximum value of the function is at n = 150, we will find the second derivative A(n).

`A''_(n)=-0.02`

Which shows A"(n) < 0

Therefore, A(n) has the maximum value at n = 150.

Therefore, number of vines that should be planted are 150.