In a rectangular box of 14x32 cm there will be a box without a lid. Determine the length of the side so that its volume is the maximum

Answer: let x = height of the box $$ V\left(x\right)=x\left(14-2x\right)\left(32-2x\right) $$ $$ V\left(x\right)=4x^3-92x^2+448x $$ $$ V^{\prime}\left(x\right)=12x^2-184x+448 $$ $$ 12x^2-184x+448=0 $$ $$ x=3.04 $$ The height of the box that could give the maximum volume is 3.04 cm