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

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

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

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



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

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

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

    Hence: .$\displaystyle 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 $\displaystyle x$
    For $\displaystyle x$ trees, the cost is: .$\displaystyle 5x$ dollars.

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

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




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

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

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


    Therefore, 105 trees should be planted for maximum profit.

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