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 planted
To 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 = 0
0.02n = 3
n = [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) < 0
Therefore, A(n) has the maximum value at n = 150.
Therefore, number of vines that should be planted are 150.
