Question

Please help with questions 6 and 7.

The exponential Eiffel Tower 501 Guided Project 72: The exponential Eiffel Tower Topics: Integration, center of mass Completed just one month before the opening of the 1889 Exposition Universelle (World's Fair) in Paris, the Eiffel Tower is one of the most recognizable landmarks in the world. It rises 300 meters from a 100 square base to a 10-meter-square observation deck. Surprisingly the project's chief engine tr no detailed structural analysis that explained the design of the tower. Recent investigations of Eiffel's notes and communications at the time the tower was built have led to a plausible model that gives a good fit to the tower's shape left In designing the tower, Eiffel's primary concern was the effect of wind loading on a free-standing structure of this size. An analysis of the forces on the tower led to the following principle that we call the Eiffel property (Figure 1): A At any height on the tower AA, the lines tangent to the tower (corresponding to the supporting forces of the tower) must pass through the center of mass of that part of the tower above AA (corresponding to the point at which a horizontal wind acts). t turns out that exponential functions have the Eiffel property very nearly, which means they give a very good description of the shape of the Eiffel Tower. Figure 1 Consider the functions y fx) =e and y = g(x) = -e"", where c is a positive real number (Figure 2). In this position, the graphs look like the Eiffel To let R be the region between the graphs on the interval [a, b]. We show that the lines tangent to the curves at x = a intersect at the center of mass (centroid) of R. x= b on the x-axis and two points x= a and tangent Hine y=f(x) =e" R y-g(x )--e" tangent line Figure 2 1. Find equations of the lines tangent to the curves y = fix) and y= g(x) atx = a. Then show that they intersect on the x-axis at the point (a 1/c, 0). Guided Projects 502 2. Explain why the center of mass of R is on the x-axis. Recall that the x-coordinate of the center of mass is To simplify matters, first assume a = 0. Then use that fact that ix)= -g(x) to show that e(bc+1) c(e-1) Now generalize Step 2 by letting a be any real number with 0s a