A rectangular piece of cardboard measuring 8 in by 10 in is to be made into a box by cutting equal size squares from each corner and folding up the sides.

Answer 1

the volume = V = x(8-2x)(10-2x) = x(4x^2-36x +80) = 4x^3 -36x^2 + 80x

If you want maximum area find the derivative of V and set = 0

V' = 12x^2 -72x + 80 = 0

6x^2 -36x + 40 = 0

x^2 -6x = -20/3

(x-3)^2 = 9-20/3 = 7/3

x = 3 + or - sqr(7/3)= 3 plus or minus 1.53 = 4.53 or 1.47. 4.53 is impossible. 1.47 gives max V

V = 1.47(8-2.94)(10-2.94) = 52.5 cubic inches

as a rough check, consider x = 1, and x= 2

they give volume = 48 each. about 1.5 the midpoint would be the likely max for x. 1.47 is very close to 1.5