In the spring of 1999 and 2001 I incorporated a series of Mathematica projects or ``labs'' into a vector analysis course. These labs used the technology of a Computer Algebra System (CAS) to further develop a particular concept, solve an application, or bring together several disjoint ideas. By far the most popular lab explored the mathematics behind the Jefferson National Expansion Memorial, commonly called the St. Louis Arch, or just the Arch.
The Arch was constructed to celebrate the westward growth of the United States brought on by the Louisiana Purchase under President Thomas Jefferson. Construction of the Arch began February 12, 1963 and was completed on October 28, 1965. (For more historical and statistical information on the Arch see the national parks website at http://www.nps.gov/jeff/). Besides being a must-see for any tourist visiting Saint Louis, the Arch has some nice mathematical properties. The Arch has height of 630 feet and is also 630 feet from base to base. The center of the Arch follows the path of an upside-down catenary curve: $A \cosh(t/A)$. Cross-sections of the Arch are equilateral triangles, the largest of these at the base has a side length of 54.1 feet, and the smallest triangle at the top has a side length of 17 feet. In the lab, students found a parameterization of the Arch and used this parameterization to compute its surface area.
In this talk I will go over a little of the mathematics involved to solve the problem as well as the impact the project had with the students.