A box without a lid is constructed from a 40 inch by 40 inch piece of cardboard by cutting x in. squares from each corner and folding up the sides.

Question

Use a graphing calculator to approximate the values of x that produce a volume of 4500.
Note: There are 3 values of x that produce the given value but only two of them are acceptable in the context of the problem. List the two answers, to at least one decimal place, separated by commas.

x=___

Answer 1

(40-2x)(40-2x)x =4500

1600x -160x^2 +4x^3 = 4500

4x^3 -160x^2 +1600x -4500 = 0 divide by 4

x^3 -40x^2 + 400x -1125 = 0

x can't be negative, the box material can't be enlarged, by adding to length and width. x<20 or volume would be negative

0<x<20 if there is a solution

x=0 then V=0

x=20 then V=0

x=9 V= 2916-11,340 + 1260<0

x=10 V=4,000-16,000+1600 <0

there could be 2 solutions, one 0<x<10 and 2nd 10<20 but volume seems negative in those ranges

V reaches 4500 at some point when x>20