A square of side x is cut out of each corner of a 13" by 22" piece of cardboard and the sides are folded up to form an open-topped box. How big should the cut-out of square be if the volume of the box is 94in^3?
base of box will be 13-2x by 22-2x, and the height will be x , 0 < x < 6.5
x(22-2x)(13-2x) = 94
x(286 - 70x + 4x^2) = 94
4x^3 - 70x^2 + 286x - 94 = 0
You will need some sort of technology to solve this cubic since it does not
factor.
Wolfram says that a value within our domain is x = appr .36
as well as x = appr 5.72
testing x = .36
box is (13 - .72)(22 - .72)(.36) = 94.07 , ok
testing x = 5.72
box is (13 - 11.44)(22 - 11.44)(5.72) = 94.2 , ok
so both values of x work