A box is to be made by cutting squares from the corners of a 22 inch by 14 inch piece of cardboard and the volume is at least 220 cubic inches. What are the possible dimensions of the squares?

Answer 1

Let's denote the dimensions of the squares to be cut out as x inches.

After cutting out the squares, the length of the box will be 22 - 2x inches, the width will be 14 - 2x inches, and the height will be x inches.

Therefore, the volume of the box will be:
V = (22 - 2x)(14 - 2x)(x)
V = (308 - 56x - 44x + 4x^2)(x)
V = (308 - 100x + 4x^2)(x)
V = 308x - 100x^2 + 4x^3

Since the volume must be at least 220 cubic inches, we can set up the inequality:
308x - 100x^2 + 4x^3 ≥ 220
4x^3 - 100x^2 + 308x - 220 ≥ 0

By factoring the expression, we can determine the possible values of x.
By solving the inequality, we find x ≥ 2 and x ≤ 7.

Therefore, the possible dimensions of the squares are between 2 inches and 7 inches.