Abox is to be constructed from a sheet of cardboard that is 10 cm by 60 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?

Volume of a rectangular box = length x width x height

From the problem statement,
length = 60 - 2x
width = 10 - 2x
height = x

where x is the height of the box or the side of the equal squares from each corner and turning up the sides

V = (60-2x) (10-2x) (x)
V = (60 - 2x) (10x - 2x^2)
V = 600x - 120x^2 -20x^2 + 4x^3
V = 4x^3 - 100x^2 + 600x

To maximize the volume, we differentiate the expression of the volume and equate it to zero.

V = 4x^3 - 100x^2 + 600x
dV/dx = 12x^2 - 200x + 600
12x^2 - 200x + 600 = 0