an open-topped box can be made from a rectangular sheet of aluminum, with dimensions 40 cm by 25 cm, by cutting equal-sized squares from the four corners and folding up the sides.

Look at my last reply to you

http://www.jiskha.com/display.cgi?id=1395791951#1395791951.1395793571

you need to solve two inequalities:

x(40-2x)(25-2x) > 1512
x(40-2x)(25-2x) < 2176

x(40-2x)(25-2x) = 4x^3 - 130x^2 + 1000x

So, you have to solve

4x^3 - 130x^2 + 1000x - 1512 > 0
4x^3 - 130x^2 + 1000x - 2176 < 0

a little synthetic division shows that the first has a root at x=2 and the second has a root at x=4

Now you are just left with two quadratics to solve, and that's easy

There are 3 intervals in the solution, but only two are feasible given positive dimensions.

The two curves are shown here:

http://www.wolframalpha.com/input/?i=plot+y%3D4x^3+-+130x^2+%2B+1000x+-+2176+and+y%3D4x^3+-+130x^2+%2B+1000x+-+1512

what do you mean by intervals

the solution set to an inequality is not just a number, but a whole range of numbers which satisfy the inequality.

Any x between 2 and 4 is a solution to this problem, as well as in the other intervals. Graphed on the number line, we have

http://www.wolframalpha.com/input/?i=solve+4x^3+-+130x^2+%2B+1000x+-+2176+%3C+0+and+4x^3+-+130x^2+%2B+1000x+-+1512+%3E+0

the plate is only 25 cm wide so the big x interval will not work

I got results in the middle of the first two intervals though

By intervals he means BETWEEN the zeros of the two functions