Suppose 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 planted, the production of the 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.
We rewrite the function as
A(n)=(700+n)(10−0.01n)A(n)=7000−7n+10n−0.01n2A(n)=7000+3n−0.01n2
The function A is a quadratic function with a=−0.01 and b=3. Thus, its maximum value occurs when
n=−b2a=−32(−0.01)=150 vines planted
The maximum production is A(150)=7000+3(150)−0.01(150)2=7225.
It shows that 150 vines should be planned in order to have a maximum production of 7225 grapes.
Monday, December 23, 2013
College Algebra, Chapter 4, 4.1, Section 4.1, Problem 70
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