A box is to be constructed from a sheet of cardboard that is 20 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? (Round your answer to two decimal places.)

Volume of the box =

V = x (60 - 2x) (20 - 2x)

= x (1200 - 40x - 120x + 4x^2)

= 4x^3 - 160x^2 + 1200x

Finding the derivative of V:-

dV/dx = 12x^2 - 320x + 1200 = 0 for a maximum volume

this gives x = 4.51, 22.15 (22.15 cannot be value of x because it would make 20 - 2x negative.

So for a maximum volume x = 4.51

and Maximum volume = (4.51) (60-9.02) (20-9.02) = 2524.5 cm^3 Answer