2070.33 – A Rectangular Box
A rectangular box without top is formed by cutting squares from the corners of an 8 x 15 sheet of cardboard and then folding up the sides. Let be the side of the square cutouts. Then tell (a) what formula in terms of gives the volume of this box, (b) whether this a quadratic formula, and (c) what's the domain of -values relevant to this problem.
According to the diagram below, the dimensions of the box are by by
(a) The volume is .
(b) This is a cubic rather than a quadratic.
(c) The relevant values of are . If is outside these limits then the box disappears, that is, it has a length negative or zero in some direction.
Note: A standard calculus problem is to find the value of that maximizes the volume of the box. Calculus is good for finding maximums or minimums of things. You could also set the problem up on a spreadsheet to hone in on the answer.