How Many Cubic Is That Model

Cubic functions and piece-wise functions can be used to model real-world situations, allowing you to identify missing bits of information you may need to complete a project. Cubic functions are commonly used to model three-dimensional objects to allow you to identify a missing dimension or explore the result of changes to one or more dimensions.

Piece-wise functions may be used to model the interactions of multiple items each previously modeled by a simpler function.

Earlier, you were asked to create a different model describing the cardboard box.

There are a number of different possibilities, for instance:

In the case of the box in the video above: A533;(A533;)=A533;/T62; could be used to describe the area of the top of the box as a function of the volume.

In the case of the box in the lesson text: A533;(A533;)=(12W22;A533;)A533; or A533;(A533;)=(8W22;A533;)A533; could be used to find the area of a divider run the long way or the wide way across the box as a function of the length of one side of the square corner cut-outs.

Consider a situation in which a rectangular piece of cardboard is folded into a box. The folding is made possible by cutting squares out of the four corners of the cardboard.

Calculate the maximum volume possible of a box made from a sheet of cardboard 12" x 8".

The function A533;(A533;)=(12W22;2A533;)(8W22;2A533;)A533; could be used to represent the volume of the box as a function of x, the side-length of the squares cut out of the corners. If we multiply out the factors of this function we can verify that this is a cubic function:

This function could be used to find the maximum possible volume of the box. We can also analyze the graph to understand how the volume changes as a function of x.

When analyzing the function to determine the maximum volume of the box, we only look at the portion of the graph that looks "parabolic". This is because the function ceases to model the situation if x is more than 4. If we cut out {4x4} squares, we would cut out the entire short side of the cardboard rectangle, and we would not be able to make a box. Focusing then on the interval (0, 4) we can see that the volume of the box increases, and then decreases.

If we are using a graphing calculator and want to know the volume of a box with particular dimensions, we can trace on the graph, input values into the table, or take advantage of the graph being in trace mode. That is, if you press GRAPH to view the graph, and then press TRACE, you can input x values. For example, say that you wanted to cut out squares of side-length 2.5. Press TRACE, then press 2.5, then press ENTER. At the bottom of the screen you will see x = 2.5 and y = 52.5. This tells you that the volume of the box will be 52.5in3.

Using the information given in Example 2 above, calculate the size of the square corners to cut out to result specifically in a volume of 50 in3.

One way to determine the value of x is to graph the constant function y = 50, and find the point where the volume function intersects y = 50. Press Y= and enter 50 into Y2. Now press GRAPH. You should see the horizontal line y = 50 intersecting the volume function in several places. We are interested in the two intersection points in the interval (0, 4).