1. What is the Largest – Volume, Open –Top, Rectangular Box You can make from…. a sheet of paper, a piece of poster board a sheet of cardboard….

2. Making the box You can get many sheets of paper and try many different x’s. You can cut and fold repeatedly and find the x by trial and error. For your first x – let’s say x = 0.5 inch, then try 1.0 inch, and repeat as necessary to find the largest volume. 11 inches 8.5 inches x

3. Try it on the internet Try it: http://mste.illinois.edu/users/carvell/3dbox/http://mste.illinois.edu/users/carvell/3dbox/ Each time you click the “cut and fold” button, the applet folds up the box and tells you its volume and surface area. By trial and error, find the x that produces the largest volume.

4. Let’s make a chart: (Use your Graphing Calculator) X-value (in inches) Length (in inches) Width (in inches) Volume of Box L1L2L3L4 0.011.08.50.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 6.5 7.537.5 in 3 9.0 10.0 58.5 in 3 66.0in 3 63.0in 3 52.5in 3 37.5in 3 21.0in 3 6.0in 3 8.0 7.0 3.0 4.0 5.0 6.0 2.5 1.5 0.5 3.5 4.5 5.5

5. What is the maximum volume of the box according to this chart?

6. What happens if our increment gets smaller? What do we need to change? Is there a faster way to accomplish this?

7. Write the function and determine its domain.