an open rectangular box is made from a square sheet of cardboard of length & width 0.5m by removing a square from each corner and joining the cut edges . Find the maximum volume of the box
Cut out a square of size x by x on each corner. Thus the length of the box is 0.5 - 2x and the length of the box is 0.5 - 2x and the height of the box is x. Thus the volume of the box will be:
V = (0.5 - 2x)*(0.5 - 2x)*x = 4x^3 - 2x^2 + 0.25x
We want the maximum value of this function.
I presume you know Calculus? We wish to set V'(x) = 0
So
V'(x) = 12x^2 - 4x + 0.25 = 0
Thus x = 1/4 and x = 1/12.
Obviously the x = 1/4 solution is ridiculous because we would be cutting the entire length of the board off in cutting the corners. So the only solution is x = 1/12.
Thus V(1/12) = 1/108 = 0.009289 m^3
-Dan