## The Box Factory

On this web page you will investigate the different ways to construct an open-top box from a sheet of paper. This exploration will reinforce the concepts in Lesson 7.6 of Discovering Advanced Algebra: An Investigative Approach.

### Sketch

This sketch shows a green sheet of paper from which you can make an open-top box by cutting out congruent squares from each corner. You can drag points W, L, and x to adjust the width, W ,of the paper, the length, L, of the paper, and the edge length, x, of the square. You'll also see a picture of the box formed from the paper.

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### Investigate

1. Adjust the width and length of the paper so that it is 16 units by 20 units. Make a table with several different values of x and the corresponding volumes.
2. Can you write functions that give the width of the box and the length of the box in terms of x?
3. Write a function that gives the volume of the box in terms of x.
4. For what value of x does the box have the largest volume? The smallest volume?

### Sketch

This sketch is similar to the first sketch, but now it also includes a graph that shows the volume of the box plotted as a function of the length, x. As you change W and L, the graph will change. The graph shows the volume of the box as a function of the box height x. When you change x, you will see the point on the graph move.

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### Investigate

1. What do you think is the degree of the function shown in the graph? Give some reasons to support your answer.
2. How can you find the box with the largest volume for a given height and width? Explain.
3. Change W and L. How do these values affect the graph of the volume function?
4. For H = 16 and L = 20, write an expression that gives the volume as a function of the square length, x. Can you factor this polynomial expression? What are the three roots of this polynomial? How can you interpret them in terms of this problem? What domain of x makes sense for modeling this situation?