Optimization problem

Question

An open box is made from a 8-by-6-inch rectangular piece of cardboard by cutting squares from each corner and folding up the sides. If x represents the side lengths of the squares, find the function giving the volume V(x) of the box in terms of x and also find the height of the box.

Answer 1

V(x) = (8 - 2x)(6 - 2x)x = 4x³ - 28x² + 48x; (0 < x < 3)

V'(x) = 12x² - 56x + 48 = 0; 3x² - 14x + 12 = 0; x = (7 ± √49 - 36)/3 = (7 ±√13)/3

V''(x) = 24x - 56;

v''((7 + √13)/3) = 20.844 > 0, at x = (7 + √13)/3 we have min

V''((7 - √13)/3) = - 28.84 < 0; at x = (7 - √13)/3 = 1.13 in we have max volume

and height h = x = (7 - √13)/3 = 1.13 inch