A rectangular box with no top is formed by cutting squares from the corners of an
8 by 15 sheet of cardboard and then folding up the sides. If the square cutouts are each x by x, then (a) what formula (in terms of x) represents the volume of the resulting box? (b) Is this a quadratic function? (c) What is the domain of x-values that are relevant to this problem?
The dimensions of the box are x by 8 – 2x by 15 – 2x.
- The volume V = x(8 – 2x)(15 – 2x) = 120x – 46x2 + 4x3
- This is a cubic function (as we’d expect), not a quadratic. (Surface area is a quadratic.)
- The domain of x is 0 < x < 4. If x = 0 there’s no box (OK, a degenerate box), and if x = 4 then 8 – 2x = 0 and that’s no good either.