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?
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.