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(a) If a farmer plants x units of wheat in his field, 0 ≤ x ≤ 144, the yield will be 12x − (x^2/12) units.

How much wheat should he plant for the maximum yield?

(b) In the problem above, it costs the farmer $108 for each unit of wheat he plants, and he is able to sell each unit he harvests for $72. How much should he plant to maximize his profit?

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2 answers
  1. y = 12 x - x^2/12
    12 y = 144 x - x^2

    x^2 -144 x = -12 y
    where is the vertex?

    x^2 - 144 x + 5184 = -12 y + 5184

    (x-72)^2 = -12 ( y - 32)
    max yield of 32 at x = 72

    p = -108 x + 72 (12 x -x^2/12)
    p = -108 x + 864 x - 6 x^2
    p = -6 x^2 + 756 x
    where is the vertex?
    6 x^2 - 756 x = -p
    x^2 - 126 x = -p/6
    x^2 - 126 x + 3969 = -p/6 + 3969
    (x-63)^2 = -(1/6) (p - 23814)
    max profit of $23,814 at 63 units planted

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  2. Thank you!

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