Chapter 2
Models of Growth: Rates of Change
Project 2: Building a Box
Preparation
For this project you will need a printer, and you may find it useful to have
- a pair of scissors,
- a ruler marked in inches,
- one or more partners so you can work as a group,
- several copies of single sheets of paper marked with half-inch grids.
These are all optional except the last. At the right we show in miniature a grid that you can print on \(8.5 \times 11\) inch paper. Click on the image (or the word "grid") to open the file. Be sure to set scaling for your printer to "none". The printed image should have half-inch margins, and every square should be one half-inch on a side. You may print this file as often as necessary.
Procedure
With each sheet of paper, you can make an open-top box by cutting out (or tearing off) equal squares at each corner. We illustrate that in the following figures, with \(x=1.5\) inch squares removed from each corner.
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- Think about the possible values for \(x.\) What will the box look like if \(x\) is small? If \(x\) is large? Are there limits on the values of \(x\)? Make a conjecture about what value of \(x\) will yield the maximum volume, and write it down.
- Cut off or tear off squares from several grid sheets and make several open-top boxes. For each box, write on the grid sheet
- the value of \(x\),
- the length of the base,
- the width of the base,
- the height of the box, and
- the volume of the box.
- Make a table of values containing \(x\) and the volume of the box corresponding to \(x.\) On a piece of graph paper, graph the values you have of volume vs. \(x.\) Where does the maximum volume seem to occur? How good was your conjecture? Is the maximum volume achieved with a wide, shallow box or a tall one with a small base?
- Find a formula for a function \(V(x)\) that expresses volume as a function of \(x\) for any value of \(x\) – not just the ones you happen to have chosen. Use your graphing tool (or ours – click button below) to graph your function. How does the graph of this function compare with the graph you drew by hand? Explain.
