Activity 3.5.4.
A piece of cardboard that is \(10 \times 15\) (each measured in inches) is being made into a box without a top. To do so, squares are cut from each corner of the box and the remaining sides are folded up. If the box needs to be at least 1 inch deep and no more than 3 inches deep, what is the maximum possible volume of the box? what is the minimum volume? Justify your answers using calculus.
(a)
Draw a labeled diagram that shows the given information. What variable should we introduce to represent the choice we make in creating the box? Label the diagram appropriately with the variable, and write a sentence to state what the variable represents.
(b)
Determine a formula for the function \(V\) (that depends on the variable in (a)) that tells us the volume of the box.
(c)
What is the domain of the function \(V\text{?}\) That is, what values of \(x\) make sense for input? Are there additional restrictions provided in the problem?
(d)
Determine all critical numbers of the function \(V\text{.}\)
(e)
Evaluate \(V\) at each of the endpoints of the domain and at any critical numbers that lie in the domain.
(f)
What is the maximum possible volume of the box? the minimum?
