A farmer finds that if she plants 55 trees per acre, each tree will yield 25 bushels of fruit. She estimates that for each additional tree p

Question

A farmer finds that if she plants 55 trees per acre, each tree will yield 25 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. How many trees should she plant per acre to maximize her harvest

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Delwyn 3 years 2021-08-26T01:38:10+00:00 1 Answers 13 views 0

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  1. Nick
    0
    2021-08-26T01:39:28+00:00
    Reply

    Answer:

    31 trees per acre will maximize the harvest

    Step-by-step explanation:

    Given

    Plant \to 55 trees/acre

    Yield \to 25 bushels

    x \to trees

    Required

    Number of trees to maximize harvest

    From the question, we understand that:

    Yield will decrease by 4 i.e. 25 – 4x

    For every additional tree planted, i.e. 55 + x

    So, the function is:

    F(x) = (25- 4x)*(55+x)

    Open bracket

    F(x) = 25 * 55 -4x * 55 + 25 * x -4x*x

    F(x) = 1375 - 220x + 25x -4x^2

    F(x) = 1375 -195x -4x^2

    Rewrite as:

    F(x) = -4x^2 -195x +1375

    The maximum of a quadratic function is calculated as:

    Max = -\frac{b}{2a}

    In the above equation:

    a = -4; b =-195; c = 1375

    So:

    x = -\frac{-195}{2 * -4}

    x = -\frac{195}{8}

    x = -24.375

    Recall that the number of trees to be planted is: 55 + x

    So, we have:

    Trees = 55+x

    Trees = 55-24.375

    Trees = 30.625

    Approximate

    Trees = 31

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