Find the dimensions of the resulting box that has the largest volume.

Question

 An open box is to be made out of a 6-inch by 20-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Dimensions of the bottom of the box: 

Height of the box: 

Answer 1

Let x = length of a side of the square

Volume of box = V = (height)(length)(width) = x(20 - 2x)(6 - 2x), where 0 < x < 3.

V = x(4x² - 52x + 120) = 4x³ - 52x² + 120x

dV/dx = 12x² - 104x + 120 = 4(3x² - 26x + 30)

dV/dx = 0 when x = [26 ± √316] / 6 = 7.3 or 1.37

Since 0 < x < 3, x = 1.37

When 0 < x < 1.37, dV/dx > 0. So, V is increasing.

When 1.37 < x < 3, dV/dx < 0. So, V is decreasing

Therefore, the relative and absolute maximum volume occurs when x = height = 1.37"

Dimensions of base: length = 20 - 2x = 17.26" and width = 6 - 2x = 3.26"