We are given a rectangular piece of material measuring 12 inches by 30 inches. Identical square cuts are made at the corners, and the resulting flaps are folded up to make an open-top box. The base of the box has an area of 208 square inches.

(See attachment.)

The width of the bottom of the box is 12-2x inches. Notice that we removed x inches from both ends. Similarly, the length is 30-2x inches. The area is found by taking the product of the length and the width, so `A=(12-2x)(30-2x)`.

We know the value of A, since we are given the area of the base to be 208 square inches. Substituting, we get

`(12-2x)(30-2x)=208`.

Multiplying the binomials on the left using the distributive property, we get

`360-24x-60x+4x^2=208`.

Combining like terms and moving all terms to the left side, we get

`4x^2-84x+152=0`.

Factor out the common factor of 4:

`4(x^2-21x-38)=0`.

By the zero-product property, the quadratic must be zero. We can solve the quadratic by factoring (if it factors over the rationals) or using completing the square or the quadratic formula.

`4(x-19)(x-2)=0` ; it factors. Again applying the zero-product property (if ab=0, then a=0, b=0, or both a and b are zero), we see that either x=19 or x=2.

`x ne 19`, since for this application, we would be removing more material than we started with, so** x=2.**

Checking: 12-2(2)=8, 30-2(2)=26, and 8(26)=208, as required.

**Further Reading**