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

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