91Ó°ÊÓ

ROMAN COLISEUM The Roman Coliseum is an elliptical amphitheater measuring approximately \(188\) meters long and \(156\) meters wide. (a) Find an equation to model the coliseum that is of the form \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\). (b) Find a polar equation to model the coliseum. (Assume and \(e\approx 0.5581\) and \(p\approx 115.98\).) (c) Use a graphing utility to graph the equations you found in parts (a) and (b). Are the graphs the same? Why or why not? (d) In part (c), did you prefer graphing the rectangular equation or the polar equation? Explain.

In Exercises 65-68, find an equation of the tangent line to the parabola at the given point, and find the \(x\)-intercept of the line. \(y=-2x^2, (-1, -2)\)

SUSPENSION BRIDGE Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table by finding the height \(y\) of the suspension cables over the roadway at a distance of \(x\) meters from the center of the bridge.

ROAD DESIGN Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure). (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.) (b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?

WRITING Explain how the central rectangle of a hyperbola can be used to sketch its asymptotes.

PROJECTILE MOTION In Exercises 81 and 82, consider the path of a projectile projected horizontally with a velocity of \(v\) feet per second at a height of \(s\) feet, where the model for the path is \(x^2 = -\dfrac{v^2}{16}(y-2)\). In this model (in which air resistance is disregarded), \(y\) is the height (in feet) of the projectile and \(x\) is the horizontal distance (in feet) the projectile travels. A cargo plane is flying at an altitude of 30,000 feet and a speed of 540 miles per hour. A supply crate is dropped from the plane. How many \(\textit{feet}\) will the crate travel horizontally before it hits the ground?