Looking at the picture they gave us, we can see that 2X's plus Z has got to equal the shorter side or the 30. So if we solve for Z, we'd get Z equal 30 -, 2 X. Looking at the longer side, we can see we have X&Y and another X&Y. So two X + 2 Y equal 48. Or if we solve for Y, we'd get 24 - X. The volume of a box is X * y * Z, So if I take X and instead of Yi put in 24 - X and instead of ZI put in 30 - 2 X, I'm going to distribute in foil and combine like terms so we get the volume equaling 720X minus 78 X squared plus 2X cubed. Taking the first derivative and setting it equal to 0, we'd have 0 equaling 720 - 156 X +6 X squared. I realize all of those numbers are evenly divisible by 6, so I'd have 0 equaling X ^2 - 20 six X + 120 as an equivalent equation. That one's much easier to factor X -, 20, X -6 X can't equal 20 because 20 would be too big. If I was trying to take two 20s away from thirty or two 20s away from the 48, that wouldn't leave enough for the other side. So X has to be 6, Y is going to be 24 - 6 or 18. Z is going to be 30 - 2 * 6 or 18 again. So 6 inches times 18 inches times 18 inches. If we'd actually asked for the volume, we would have just multiplied 6 * 18 * 18.