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 1078355: 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 _______________
oranges per acre, plant___________ trees on each acre.
Answer by ikleyn(53937) About Me  (Show Source):
You can put this solution on YOUR website!
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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 = -b%2F%282a%29 = -930%2F%282%2A%28-15%29%29 = 930%2F30 = 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)) = 21*(630+15*11) = 16695.


Answer.  31 trees per acre provide the maximal number of oranges of 16695.

*** Solved ***




Plot y = t*(630 - 15*(t-20))


On finding the maximum of a quadratic function and associated solved problems see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola

    - Using quadratic functions to solve problems on maximizing revenue/profit


Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".