Chapter Project 753 7. Find all the solutions of the equation sin 2 0.5. θ ( ) = 8. Find a polar equation for the line containing the origin that makes an angle of 30° with the positive x-axis. 9. Find a polar equation for the circle with center at the point 0, 4 ( ) and radius 4. Graph this circle. 10. What is the domain of the function f x x x 3 sin cos ? ( ) = + 11. Solve the equation cot 2 1, θ ( ) = where 0 90 . θ ° < < ° 12. Find the rectangular equation of the plane curve x t t y t t t 5 tan 5sec 2 2 2 π π ( ) ( ) = = − < < is an astronomical unit? What is it equivalent to in miles? In kilometers? What is the orbital period of Comet Hale-Bopp? When will it next be visible from Earth? How close does it come to Earth? 2. Find a model for the orbit of Comet Hale-Bopp around the Sun. Use the x-axis as the major axis. 3. Comet Hale-Bopp has an orbit that is roughly perpendicular to that of Earth. Find a model for the orbit of Earth using the y-axis as the major axis. 4. Use a graphing utility or some other graphing technology to graph the paths of the orbits. Based on the graphs, do the paths of the orbits intersect? Does this mean that Comet Hale-Bopp will collide with Earth? Chapter Project Comet Hale-Bopp The orbits of planets and some comets about the Sun are ellipses, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. See the figure. Major axis Sun Center Aphelion Mean distance Perihelion 1. Research the history of Comet Hale-Bopp on the Internet. In particular, determine the aphelion and perihelion. Often these values are given in terms of astronomical units. What Internet-based Project Credit: MarcelClemens/Shutterstock
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