Optimization: Constructing a Box Lesson Information Written by Jonathan Schweig and Shira Sand Subject: Pre-Calculus Calculus Algebra Topic: Functions Overview: This lesson walks students through a classic optimization problem involving building a (lidless) box from a sheet of paper. The lesson uses a worksheet in The Geometer s Sketchpad. Technology: Geometry Software Level: Difficult Activity Structure: Self-Guided Problem Solving Duration of Activity: Whole Class Period Multiple Classes Learning Objectives: Students will be able to look for patterns and determine the volume of the largest constructible box. Students will be able to use this knowledge to solve similar problems Students will gain an understanding of the problem solving process used to solve optimization type problems Materials: The Geometer s Sketchpad Software GSP Worksheet entitled Box Paper Pencil Worksheet
Problem: Determine the dimensions of a box of maximum volume that can be made from a piece of material 16 cm X 20 cm. The box is to be made by cutting square pieces from the corners and folding up the sides. The box will not have a top. Exploration: 1. Click on Make Cut button in order to cut square pieces from the corners of the material. 2. Drag the point labeled Drag Here to change the length of the cut. Observe how the dimensions of the box change as you change the length of the cut. In what way do these dimensions relate to the length of the cut? 3. What cut do you think will create the desired box? 4. Click on Show Dimension Table. This table gives information about the dimensions of the box for a particular length cut. Create a table (remember: in order to store data, double click on the table) for the following cut length values: 0.5, 1, 1.5, 2, 2.5, and 3. Record your table below.
5. What do you notice about: a. the relationship of the box s height to the length of the cut? b. the relationship the box s length to the length of the cut? c. the relationship of the box s width and the length of the cut? 6. Click on Hide Dimension Table. Now, click on Show Volume Table. This table gives information about the volume of the box for a particular length cut. Create a table using the cut lengths from question 5 as well as your guess from question 4. Record your table below. 7. Is the box you proposed in question 3 the box with the largest volume on your table?
8. Plot a rough sketch of volume vs. cut length on the coordinate axes below. 9. What type of mathematical relationship does this resemble? 10. Use the Graphs tab on the bottom of your window to move to the next page of the workbook. Click on Change dimensions. Watch the trace of the plotted point as the dimensions of the box change. You can click on Change dimensions again in order to stop the action. What type of mathematical relationship does this resemble? Does this match your answer from question 9? 11. What point on the graph represents the maximum volume of the box?
12. Drag the point to the cut length you guessed in question 3. Is the plotted point at the maximum value of your graph? 13. Click on Move to maximum volume. How far was your hypothesized length from the actual length of the desired cut? 14. Describe the shape of the box with the maximum volume.
Generalization: 1. Consider a rectangular piece of material with length l and width w. Squares are cut out of each corner with size x, and the sides are folded up to create a box. a. Describe the height of the box in terms of x. b. Describe the length of the box in terms of x. c. Describe the width of the box in terms of x. 2. Write a function of x representing the volume of the box. 3. What type of function is this? What do you know about the behavior of this type of function? 4. What is the domain of your function? Hint: refer to the illustration if necessary, consider all possible cut lengths. 5. Were your predictions in questions 10 and 11 of the exploration correct? If not, explain what caused you to come to an incorrect conclusion. 6. Describe a method for finding the maximum value for this type of function.
Extensions: 1. A vendor sells Valentine candy boxes to 500 customers at the price of $10.00 a box. He discovers that for each $0.25 increase in the price of a Valentine candy box, he will lose two customers. For what price should he sell each box to bring in the greatest total gross income? 2. The owner of several hotels is converting them, one by one, into rental units. In each building, all the units can be rented if the rent is set at $600 per unit. In each building, for each $20 that the monthly rent is raised, one of the units in that building becomes vacant. Each occupied unit requires $100 in service each month. The owner would like to know how much rent should be charged per unit in a building with 35 units in order to maximize the cash flow from that building.