Abox is to be constructed from a sheet of cardboard that is 10 cm by 50 cm by cutting out squares of length x by x from each corner and bending up the sides. what is the maximum volume this box could have?

If x is the side of the squares being cut out of the sheet, then when the sides are bent up, the box that results has dimensions x, 10 - 2x, and 50 - 2x. The volume is



The derivative is

Set that equal to 0 and solve for x (answers are weird!).



Use a calculator to plug those into the volume function, then pick the largest.