Question

A square sheet of cardboard with each side a centimeters is to be used to make an open-top box by cutting a small square of cardboard from each of the corners and bending up the sides. What is the side length of the small squares if the box is to have as large a volume as possible?

Answer 1

Let the side of each of the squares to be cut out be x cm
then the base is (a-2x), and the height of the box is x cm, where a is a constant.

Vol = x(a - 2x)^2 = x(a^2 - 4ax + 4x^2)
= a^2 x -4ax^2 + 4x^3
d(Vol)/dx = a^2 -8ax + 12x^2
= 0 for a max of Vol

12x^2 - 8ax + a^2
x = (8a ± √(64a^2 - 4(12)(a^2) ) )/24
= (8a ± √16a^2)/24
= (8a ± 4a)/24
= a/2 or x = a/6 , clearly a/2 will not work giving us a zero base

x = a/2 or (1/2)a

check my algebra, I did not check it

Answer 2

A square sheet of cardboard with each side centimeters is to be used to make an open-top box by cutting a small square of cardboard from each of the corners and bending up the sides. What is the side length of the small squares if the box is to have as large a volume as possible?