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23 May, 21:55

Find the greatest possible value of the product xy, given that x and y are both positive and x + 2y = 30

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  1. 23 May, 22:05
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    Greatest value of xy

    x + 2y = 30, x = 30 - 2y

    xy = (30-2y) y = 30y-2y^2

    so if - 2y^2+30y = 100, then we can take out a (-2), so: y^2-15y=-50, y^2-15y+50=0

    (y-5) (y-10) = 0

    y-5=0, y=5

    y-10=0, y=10

    Now we can "plug n play": plug each into x = 30-2y

    x = 30-2 (5) = 30-10 = 20

    x = 30-2 (10) = 30-20 = 10

    So if we make x=20, y=5 ... xy = 100

    And if x=10, y=10 ... xy = 100

    Now we need to try other number variations to see if they can make xy > 100

    if y=6, then x = 30-2 (6) = 30-12 = 18

    xy = (18) (6) = 108. Ooh! there's a better choice, but let's keep going

    if y=7, then x = 30-2 (7) = 16, xy = 7*16 = 112

    Even higher! But let's check others

    If y=8, then x = 30-16=14, so xy = 112

    But from here, the higher y gets the product of xy starts to decrease again

    So the greatest possible value comes from 7*16 and 8*14, in which x*y was the maximum it could be at 112
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