Peter R. answered 04/03/23
Experienced Instructor in Prealgebra, Algebra I and II, SAT/ACT Math.
A piece of cardboard is 12" x 18" and you cut squares of x" from each of the four corners. The length of the resulting open box will be 18 - 2x inches; the width will be 12 - 2x inches; the height will be x inches.
So the volume that you need to maximize is (18 - 2x)(12 - 2x)(x)
If you expand that you'll get V = 4x3 - 60x2 + 216x.
Can find max value of V by taking 1st derivative and setting equal to zero and solve for x, or plot the graph on Desmos Graphing Calculator.