Something in the Way of Calculus

        We have a rectangular sheet of tinfoil whose dimensions are 32 centimetres by 20 centimetres.

 Equal squares are cut out at each of the corners.

       Can you find the maximum volume of a wooden box which the base and sides of the box?

Answer:    1152 cubic centimetres.

Let us assume v is the volume o f the box, and x is the side of the squares cut out.

Then v = (32-2x) (20 – 2x) x

Or    v= 640x – 104x² + 4x³

 Dv / dx = 640 – 208x + 12x²

 For a maximum volume dv/dx = 0

Or 3x² - 52x + 160 = 0

Or (3x - 40) (x – 4) = 0

Or x= 4 or x= 40/3

The real volume of x can only be 4. Therefore the maximum volume of the box is 24x 12 x a4 = 1152 cubic centimetres.