Preview Activity 5.3.1.
A piece of cardboard that is \(12 \times 18\) (each measured in inches) is being made into a box without a top. To do so, squares are cut from each corner of the cardboard and the remaining sides are folded up.
(a)
Let \(x\) be the side length of the squares being cut from the corners of the cardboard. Draw a labeled diagram that shows the given information and the variable being used.
(b)
Determine a formula for the function \(V\) whose output is the volume of the box that results from a square of size \(x \times x\) being cut from each corner of the cardboard.
(c)
What familiar kind of function is \(V\text{?}\)
(d)
If we start with a small positive value for \(x\) and let that value get larger and larger, what is the first value of \(x\)we encounter that makes it impossible to remove \(x \times x\) squares from the cardboard and still form a box?
(e)
What are the zeros of \(V\text{?}\) What is the domain of the model \(V\) in the context of the rectangular box?
