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Math Help - revenue

  1. #1
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    Post revenue

    Please Help I am completely confused on how to do this. Thanks

    REVENUE
    : An apple orchard produces annual revenue of $60 per tree when planted with 100 trees. Because of overcrowding, the annual revenue per tree is reduced by $0.50 for each additional tree planted.
    Assuming , write an equation for the revenue produced by trees:

    How many trees should be planted to maximize the revenue from the orchard?
    Now suppose that the cost of maintaining each tree is $5 per year.
    Write the profit function in terms of :

    How many trees should be planted to maximize the profit from the orchard?
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  2. #2
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    This is an example of an optimisation problem using linear programming. Have a look at this and see if it makes sense:

    [html]http://en.wikipedia.org/wiki/Linear_programming[/html]
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  3. #3
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    Hello, boondocksaint202!

    An apple orchard produces annual revenue of $60 per tree when planted with 100 trees.
    Because of overcrowding, the annual revenue per tree is reduced by $0.50
    for each additional tree planted.

    (a) Assuming x \geq 100, write an equation for the revenue produced by x trees.
    For every tree over 100, the revenue per tree is reduced by \$\frac{1}{2}

    The overage is x - 100.
    This reduced the revenue to: . 60 - \frac{1}{2}(x-100) dollars per tree.

    The Revenue is: .[no. of trees ] x [revenue per tree]

    . . R \;=\;x\bigg[60 - \frac{1}{2}(x - 100)\bigg] \quad\Rightarrow\quad \boxed{R \;=\;110x - \frac{1}{2}x^2}



    (b) How many trees should be planted to maximize the revenue from the orchard?
    The Revenue function is: . R  \;=\;110x - \frac{1}{2}x^2

    This is a down-opening parabola; its maximum is at its vertex.
    . . The vertex is at: . x \:=\:\frac{-b}{2a}

    We have: . a = \text{-}\frac{1}{2},\;b = 110

    Hence: . x \:=\:\frac{\text{-}110}{2\left(\text{-}\frac{1}{2}\right)} \:=\:110


    Therefore, 110 trees should be planted for maximum revenue.




    (c) Suppose that the cost of maintaining each tree is $5 per year.
    . . Write the profit function in terms of x
    For x trees, the cost is: . 5x dollars.

    . . \text{Profit} \;=\; \text{Revenue} - \text{Cost}

    . . . P \;=\;\left(110x - \frac{1}{2}x^2\right) - 5x \quad\Rightarrow\quad\boxed{ P\;=\;105x - \frac{1}{2}x^2}




    (d) How many trees should be planted to maximize the profit from the orchard?
    The profit function is: . P\;=\;105x - \frac{1}{2}x^2

    This is a down-opening parabola ... with its maximum at its vertex.

    The vertex is at: . x \:=\:\frac{\text{-}105}{2\left(\text{-}\frac{1}{2}\right)} \;=\;105


    Therefore, 105 trees should be planted for maximum profit.

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