Question

A candy box is made from a piece of cardboard that measures 33 by 18 inches. Squares of equal size will be cut out of each corner. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum​ volume?

Answer 1

Given:  cardboard that measures 33 by 18 inches.

To find: What size square should be cut from each corner?

Solution:

• As we have given the size of the cardboard, so lets consider the length of the square be 'x'.

• So, length, width and height will be:

                Length = 33 − 2x

                 Width = 18 − 2x

                 Height = x

• So now, the volume will be:

                 Volume = (33 − 2x) x (18 − 2x) x (x)

• After calculating volume comes out to be:

                 V = (594 − 66x − 36x + 4x²) (x)  

                 V = 4x³ − 102x² + 594x

• Now, we can use differentiation to equate it to zero.
• So differentiate it with respect to x, we get

                 dV/dx = 12x² − 204x + 594

                 12x² − 204x + 594 = 0

• Dividing the equation first by 2 and then 3 we get:

                 2x² − 34x + 99 = 0

• So, after solving this, x comes out to be:

                  x = 13.267 and x = 3.73

Answer:

              So size of square should be 3.73 inches.