A box is made from a sheet of cardboard by cutting squares f

A box is made from a sheet of cardboard by cutting squares from the comers and folding up the sides. If the cardboard is 18 inches by 24 inches, find: The maximum possible volume of the box. The length of the side of the square to be cut out if the volume is to be 275 cubic inches.

Solution

Let the length of square to be cut is x

Length of box aftercutting from corners:

18 -2x

width of box after cutting from corners:

24 -2x

Volume = length*width*height = (18-2x)(24-2x)x = x( 432 -36x -48x +4x^2) = 4x^3 -84x^2 +432x

a) Max. volume we get at dV/dx =0 ; 12x^2 - 168x+432 =0

solve the quadratic : x = 10.69 inch , x = 3.39 inch

Maximum volume at x =3.39 inch = 654.97 inch^3

b) volume = 275 inch^3

4x^3 -84x^2 +432x = 275

solve the cubic equation : we get x = 13.23 ; x =0.74 ; x= 7.03