## The Calculus Whiz Who Owned a Box Company

### Linda Knowles

#### Description

Students develop an understanding of the relationship between volume and surface area. They then construct a box out of a piece of paper that maximizes volume using a table, by graphing and calculus techniques.

#### Objectives

Adds, subtracts, multiplies, and divides real numbers, including square roots and exponents using appropriate methods of computing (mental mathematics, paper-and-pencil, calculator).

Uses concrete and graphic models to derive formulas for finding perimeter, area, surface area, circumference, and volume of two- and three- dimensional shapes including rectangular solids, cylinders, cones and pyramids.

Describes, analyzes and generalizes relationships, patterns, and functions using words, symbols, variables, tables and graphs.

#### Materials

-Pencil
-Paper
-Graphing Calculator
-Ruler
-Graph paper
-Card stock or construction paper
-Scissors
-Tape
-Calculus Whiz Box Company worksheet (see Associated File)

#### Preparations

1. Locate a ruler, scissors, graphing calculator, graph paper and an 8.5x11 inch piece of paper or card stock for each group.
2. Duplicate one copy of File Document #1 for each student.

#### Procedures

This lesson is part of a series of lessons entitled "The Calculus Whiz Who …"

Prior knowledge required: Students must be able to find the volume and surface area of a rectangular solid. They must also be able to write the equation for the volume of a solid, graph this equation on a graphing calculator and find the maximum value of this function both on the graph and by using calculus (differentiation and critical values, etc.).

1. Gain the students' attention by telling them about the calculus whiz who was quite a math nerd (and proud of it). He owned a box company and after observing that the same size piece of cardboard could be made into different shaped boxes, each with the same surface area but different volumes. He trusted that his beloved calculus would help him create a box out of a set size of cardboard that would have maximum volume.

2. Review on the board or overhead how to find the volume and surface area of a right rectangular solid. For surface area, pretend that the box has no lid.

3. Place students in groups of three or four.

4. Pass out to each student the worksheet entitled "Calculus Whiz Box Company" (File Document #1). All during this project they should work as a group but each student should fill in his/her own worksheet.

5. Give students a few minutes to find the volume and surface area of each box on this worksheet in part A.

6. Discuss the conclusion they should have made about the results in part A.

7. Pass out to each group a piece of card stock or construction paper (8.5 by 11 inches) and tell them that each group will work together to construct a box with no lid that has the maximum volume. This will be done by cutting out squares from the four corners of the sheet of paper and turning up the sides to form a box as illustrated in part B.

8. Tell them to fill out the chart in part B on the worksheet to justify their conclusion about the size of the squares to be cut out at each corner to maximize volume.

9. Be sure to discuss why it makes no sense in this problem to go beyond 4.25 inches for the size of the squares to be cut out. (0 to 4.25 is the domain of x.)

10. Circulate around the room to help each group as needed.

11. Have them actually cut out the squares, turn up and tape the sides to form a box. Write group members' names on the bottom of the box and turn it in.

12. Give each group a few minutes to write an equation for the volume of the box in terms of x. (Part C on the worksheet) Check that they all have the correct equation before going on to #13.

13. Have them graph this equation on the graphing calculator and transfer this graph to the graph paper. Instructions are in Part D on the worksheet.

14. Tell them to find the highest point on the graph between 0 and 4.25 using the maximum key on the calculator. The y value of this point will be the maximum volume. Label this point on the graph.

15. In Part E of the worksheet, have the students differentiate the equation for the volume that they wrote in part C and find the critical values (where V'(x) = 0 or is undefined). This answer should correspond to the x-value of the highest point on the graph.

16. Fill out the chart to justify the answer using the theorem mentioned in part E.

#### Assessments

Observe the group activity and assess the papers turned in from each group (File Document #1) on a percentage basis. Staple the group papers together. A suggested scoring guide is also included.

#### Attached Files

File Attachment.     File Extension: pdf