Respuesta :
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":
[tex]A(n)=(600+n)(10-0.01n)\\A(n)= 6000-6n+10n-0.01n^2\\A(n) = 6000+4n-0.01n^2[/tex]
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: [tex]y=ax^2+bx+c[/tex] has an x-component defined by:
[tex]x_{vertex}=\frac{-b}{2a}[/tex]
then in our case, the "n" value for that vertex is: [tex]n_{vertex}= \frac{-4}{2(-0.01)} =200[/tex]
Then the additional number of vines to maximize grape production is 200
So the total should be: 600 + 200 =800
To maximize the grapes production 800 vines should be planted.
Steps to maximize the production:
- Differentiate the function for the production with respect to the
variable given in the function.
- Equate it to zero and find the value of the variable.
Given in the question,
Function representing the number of pounds of grapes produced per acre,
- A(n) = (600 + n)(10 - 0.01n)
Here, n = number of additional vines planted
Simplify the function,
A(n) = 600(10 - 0.01n) + n(10 - 0.01n)
= 6000 - 6n + 10n - 0.01n²
= -0.01n² + 4n + 6000
- Differentiate the function with respect to variable 'n',
A' = -0.02n + 4
- Equate it to zero,
A' = 0
0.02n + 4 = 0
n = 200
Additional vines to be planted = 200
Hence, total number of vines = 600 + 200 = 800
Therefore, 800 vines should be planted to maximize the grape production.
Learn more about the maximization here,
https://brainly.com/question/5765993?referrer=searchResults
