SOLUTION: Each orange tree grown in California produces 630 oranges per year if not more than 20 trees are planted per acre. For each additional tree planted per acre, the yield per tree dec
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Question 1078727: Each orange tree grown in California produces 630 oranges per year if not more than 20 trees are planted per acre. For each additional tree planted per acre, the yield per tree decreases by 15 oranges. How many trees per acre should be planted to obtain the greatest number of oranges?
To achieve a maximum yield of 16695 oranges per acre, plant 31 trees on each acre.
I got this answer, however 16695 is incorrect please help me get the right answer. Found 2 solutions by Boreal, ikleyn:Answer by Boreal(15235) (Show Source):
You can put this solution on YOUR website! Let x=each additional tree.
(630-15x)(20+x) is the equation.
Maximize 12600+330x-15x^2=0
The vertex is -b/2a=-330/-30=11 for x
31 trees, 465 oranges per tree (630-(15*11))
14415
For the number of oranges per one tree, "n", as the function of the number of trees per acre, "t", the condition gives this formula:
n = 630 - 15*(t-20).
Then the number of oranges per acre is
O(t) = n*t = t*(630 - 15*(t-20)) = 630t - 15t^2 + 300t = -15t^2 + 930t.
They ask to find the maximum of this quadratic function of t.
From the general theory, the maximum is achieved at t = = = = 31.
So, 31 tree per acre provide the maximal total number of oranges per acre.
This maximum is equal to O(31) = 31*(630-15*(31-20)) = 31*(630-15*11) = 14415.
Answer. 31 trees per acre provide the maximal number of oranges of 14415.
