# Maximize the Volume of a Box: Exploring Polynomial Functions

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

#### Summary

In this Spreadsheet Across the Curriculum activity, students will create spreadsheets to find the maximum of a polynomial function. The module addresses the following optimization problem. Given a piece of cardboard 8 inches by 10 inches on a side, and letting x represent the length of a square cut out of each of the four corners of the cardboard sheet, what value of x produces the largest volume of open-top box made by folding up the cut-up cardboard? After exploring an applet available on the Internet, the students find x by creating a spreadsheet that evaluates and graphs the volume of the box as a polynomial function of x for many values of x. Then they find x again by differentiating the function, setting the derivative equal to zero, and creating a spreadsheet to solve the resulting quadratic equation in x.

## Learning Goals

Students will:
• Explore the Three Dimensional Box Applet
• Start with a spreadsheet formula to calculate the volume of a box from a variation of length times width times depth.
• Embed the calculation as a row in a spreadsheet in which the calculation is repeated through many rows to reveal how the volume of the box varies with x, the length of the square cutout noted in the above summary.
• Examine the spreadsheet to find the x that produces the maximum volume.
• Create an x-y scatter graph volume vs. x to observe the local maximum of the polynomial function.
• Find the equivalent polynomial and from it use calculus to find the local maximum of the function.
• Apply the formula to solve a quadratic equation.
In the process the students will:
• Visualize an optimization problem and learn how to solve one numerically using a spreadsheet.
• Learn that a use for the quadratic formula appears in non-mathematics context.
• See that algebra and calculus can be used to solve the same problem.

## Context for Use

I will use this module in my classes such as Math 100 (Elementary Algebra), Math 145 (Precalculus), and Math 160 (Calculus I). It can be used in these courses as a long homework, mini-project, or in class group discussion. We can use this module in all these classes the same way or differently. For example, in calculus we can justify the results of Microsoft Excel with the traditional approach of finding maximum value of a function by using the first and second derivative tests. This module can enhance learning of elementary geometry, polynomials, functions, graphs, multiplication of polynomials, factoring polynomials, quadratic equations, quadratic formula, closed/open intervals, and more.

Additionally, we can challenge students to do the same idea of this module for non-rectangular solids, cones, sphere, and other shape solids in geometry by using Microsoft Excel to solve optimization problems.

## Description and Teaching Materials

The module is a PowerPoint presentation with embedded spreadsheets. If the embedded spreadsheets are not visible, save the PowerPoint file to disk and open it from there.

The above PowerPoint files are the student version of the module. An instructor version is available by request. The instructor version includes the completed spreadsheet. Send your request to Len Vacher (vacher@usf.edu) by filling out and submitting the Instructor Module Request Form.

## Teaching Notes and Tips

Students need to know elementary algebra, geometry, first and second derivative tests in calculus and how to use mathematics formula in Excel as spreadsheet.

## Assessment

Pre-test, post-test, and assignments are available for the assessment purpose in the module.

## References and Resources

Online educational information related to this module