(Continuation of Exercise 55.) Suppose that a flexible

Chapter 5, Problem 56

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(Continuation of Exercise 55.) Suppose that a flexible cable of uniform density is suspended between two supports, as indicated in Figure A. (Example: a telephone line between two phone poles.) Using calculus and physics, it can be shown that the curve formed is a portion of the graph of y a cosh(x/a), where a is a constant. See Figure B. (The actual value for a depends on the tension in the cable and the density of the cable.) The graph of an equation of the form y a cosh(x/a) is called a catenary (from the Latin catena, meaning chain). Figure A Figure B (a) Use a graphing utility to graph the four catenaries y a cosh(x/a) corresponding to a 0.2, 1, 3, and 5. Describe (in complete sentences) how the shape and location of the curve changes as the value of a increases. (b) The Gateway Arch in St. Louis (shown on page 336) is built in the form of an inverted catenary. The equation for the curve is given in Figure C. In this equation both x and y are measured in feet. Find the height of the Gateway Arch, then determine the width by using a graphing utility to graph the equation given in Figure C. Remark: The Gateway Arch is the tallest monument in the U.S. National Park System. Designed by architect Eero Saarinen and completed in 1965, the Gateway Arch commemorates the Louisiana Purchase, and more generally, the westward expansion of the United States over the period 18031890. Fi