7 Analyzing Accumulated Change: More Applications of Integrals

 Project 7.1: Arch Art Setting A popular historical site in Missouri is the Gateway Arch. Designed by Eero Saarinen, it is located on the original riverfront town site of St. Louis and symbolizes the city's role as gateway to the West. The stainless steel Gateway Arch (also called the St. Louis Arch) is 630 feet (192 meters) high and has an equal span. In honor of the 200th anniversary of the Louisiana Purchase, which made St. Louis a part of the United States, the city has commissioned an artist to design a work of art at the Jefferson National Expansion Memorial National Historic Site. The artist plans to construct a hill beneath the Gateway Arch, located at the Historic Site, and hang strips of mylar from the arch to the hill so as to completely fill the space. (See Figure 7.54.) The artist has asked for your help in determining the amount of mylar needed. FIGURE 7.54 Tasks If the hill is to be 30 feet tall at its highest point, find an equation for the height of the cross-section of the hill at its peak. Refer to Figure 7.54. Estimate the height of the arch in at least ten different places. Use the estimated heights to construct a model for the height of the arch. (You need not consider only the models presented in this text.) Estimate the area between the arch and the hill. The artist plans to use strips of mylar 60 inches wide. What is the minimum number of yards of mylar that the artist will need to purchase? Repeat Task 4 for strips 30 inches wide. If the 30-inch strips cost half as much as the 60-inch strips, is there any cost benefit to using one width instead of the other? If so, which width? Explain. Reporting Write a memo telling the artist the minimum amount of mylar necessary. Explain how you came to your conclusions. Include your mathematical work as an attachment.