At a certain vineyard it is found that each grape vine produ

At a certain vineyard it is found that each grape vine produces about 10 pounds of grapes in a season when about 700 vines are planted per acre. For each additional vine that is painted, the production of 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 - 0.01 n) where n is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

Solution

given total production A(n)=(700+n)(10-0.1n)

fdor local maximum or minimum dA/dn =0

differentiate with respect to n on both sides

=>(0+1)(10-0.1n) +(700+n)(0-0.1)=0

=>10-0.1n-70-0.1n=0

=>-0.2n=70-10

=>n =60/(-0.2)

=>n =-300

A(0)=(700+0)(10-0)=7000 pounds

A(-300)=(700+(-300))(10-0.1(-300))

A(-300)=(400)(10+30)

A(-300)=16000

so maximum production =16000pounds when n=-300

number of vines =700+n

number of vines =700-300

number of vines =400