An open-top box is made from a 24in x 36in cardboard piece by taking a square from each corner of the box and folding up all 4 side flaps. What size squares should be cut to get a box with max volume?

Question

If you can, please show all work and write a detailed explanation-- I would really like to understand this problem!

An open-top box is to be made from a 24 inch by 36 inch piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. What size square should be cut out of each corner to get a box with the maximum volume?

Answer 1

You cut a square of side x at the 4 corners and fold it up.

The base of the box will be a rectangle of sides 24 - 2x and 36 - 2x (you cut x on both ends of each side).

The height of the box will be x.

You want to maximize the volume: V(x) = base area . height = (24 - 2x)(36-2x)x = 4x(12-x)(18-x).

Because the sides of the box should be positive: x > 0 and x < 12 and x < 18 i.e 0 < x < 12

So expand V(x), find the first derivative, set it to zero (will give you a quadratic equation) and find the maximum location x_max on the interval 0 < x < 12. The maximal volume will be V_max = V(x_max).