Suppose you own a catering business. As part of your services, you not only prepare the food, but you also set up tables and chairs. Assume one person can sit comfortably at the edge of the table. You need to set up for 18 people. Since your tablecloths are large, tables must meet along at least one edge. (In other words you can not have single tables that seat four people)
How many tables will you need? Are there other table arrangements that are possible that would require the fewest number of tables? the greatest number of tables? If so, is there a special significance to these solutions? What happens to the perimeter when one table is added? (make sure to consider all cases) What, in mathematical terms, do the number of people and the number of tables represent?
If you have not already done so, put your data in a table. Graph various combinations of the data, identifying the various relationships.
If we allow the dimensions to be any real numbers (not just integers), what are the dimensions with the greatest area? How can we be sure that the maximum area is obtained using these dimensions?
Suppose you have a total of 24 tables. How many people can be seated around 24 tables? (The same criteria still holds) Find as many possible arrangements as you can.
What patterns do you observe? How does this problem compare with the previous problem? What is the same? What is different?
What is the largest perimeter? Could we have a larger perimeter? What about the minimum perimeter? Could we get a smaller perimeter?
If we extend our investigations to other figures in the plane (e.g. other polygons and circles), which plane figures will have the greatest area for a fixed perimeter?
Given ax^2 + bx + c, use Algebra Expressor to explore the effect on the graph if we change the value of the coefficients.