An open box is to be made from a square piece of material with a side length of 10 inches by cutting equal squares from the corners and turning up the sides. What size of a square would you cut off if the volume of the box must be 48 cubic inches.

When you cut off for squares of x side length, the area of the base of the box is [10 - 2x]^2 and the height ot the box is x, then the volumen of the box is:

(10 - 2x) ^2 * x = 48

(100 - 40x + 4x^2) * x = 48

100x - 40x^2 + 4x^3 = 48

4x^3 - 40x^2 + 100x - 48 = 0

x^3 - 10x^2 + 25x - 12 = 0

Using Ruffini you can get 3 as a solutio:

3^3 - 10 (3^2) + 25 (3) - 12 = 27 - 90 + 75 - 12 = 0

Factoring you get (x - 3) (x^2 - 7x + 4) = 0

Then, you can apply the quadratic formula to x^2 - 7x + 4 to find the other two roots.

They are x = 0.6277 and x = 6.3723.

x = 6.32723 is not valid because 10 - 2x < 0.

Then there are two real solutcions for the side of the squate: 3 and 0.6277 (althoug the second one is a very low box)