What size of square would you cut out if the volume of the box must be 48 cubic inches?

Question

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 corners and turning up the sides.

Answer 1

Draw a picture

 

After cutting out a square that is x by x from each corner, the base is 10 - 2x by 10 - 2x

 

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

48 = (100 - 40x + 4x²)x

48 = 100x - 40x² + 4x³

0 = 4x³ - 40x² + 100x - 48

 

Descartes Rule of Signs indicates three real roots, or one real root and two complex roots (this is the one I suspect)

Using the Rational Root Theorem and Synthetic Substitution,

 

I found x = 3.