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Name_____________________________ HW 90 APPLICATIONS OF CONIC SYSTEMS Date ________________ Per_______ Pre-AP Algebra 2 Do all work on your own paper. 1. A satellite in a 100-mile high circular orbit around the earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by 2 , then the satellite will have the minimum velocity necessary to escape the earth’s gravity and it will follow a parabolic path with the center of the earth as the focus. a) Find the escape velocity of the satellite. b) Find the equation of the path (assume that the radius of the earth is 4000 miles) 2. A semi-elliptical arch over a tunnel for a road through a mountain has a major axis of 100 feet and its height at the center is 30 feet. Write an equation of the ellipse, and then determine the height of the tunnel 5 feet from the edge of the tunnel. (see page 739 for a picture reference) 3. The earth moves in an elliptical orbit with the sun at one of the foci. The length of half the major axis is 92 .957 X 10 6 miles and the eccentricity is 0.017. (The eccentricity of the ellipse is the ratio c/a). Find the smallest distance (perihelion) and the greatest distance (aphelion) of the earth from the sun. 4. The path of a projectile projected horizontally with a velocity of v feet per second at a height of s feet is modeled by the equation y 16 t 2 s . [In this model air resistance is disregarded and y is the height in feet of the projectile t 2 v seconds after its release.] a) If a ball is thrown horizontally from the top of a 100-foot tower with a velocity of 32 feet per second, find the equation of the parabolic path and find how far the ball travels horizontally before striking the ground. b) A bomber flying due east at 550 miles per hour at an altitude of 42,000 feet releases a bomb. Determine how far the bomb travels horizontally before striking the ground. 5. The path followed by a baseball after it is hit can be modeled by the equation 2x2 – 800x + 1000y – 4000 = 0, measured in feet. a) Write the equation in standard form. b) What is the maximum height of the ball? c) What was the height of the ball when it was hit? 6. The main cables of a suspension bridge are ideally parabolic. The cables over a bridge that is 400 feet long are attached to towers that are 100 feet tall. The lowest point of the cable is 40 feet above the bridge. (see page 756) a) Find the coordinates of the vertex at the tops of the towers if the bridge represents the x-axis and the axis of symmetry is the y-axis. b) Find an equation that can be used to model the cables. 7. If the x-axis is placed at a height of 100 meters, the outer edge of a cooling tower can be modeled by the hyperbola x2 y2 1 , measured in meters. If the tower is 150 meters tall, find the width of the cooling tower at the top. 900 1600 See page 749) 8. The outline of a cube spinning around an axis through a pair of opposite corners contains a portion of a hyperbola, as shown in the figure. The coordinates given represent a vertex and a focus of the hyperbola for a cube that measures 1 unit on each edge. Write an equation that models this hyperbola. 9. If a large room is constructed in the shape of the upper half of an ellipsoid, it will have a unique property. Two people, one at each of the two foci, can hear each other whispering no matter what the distance between them. If the two people are 100 feet apart, and the maximum height of the ceiling is 40 feet, estimate the area of the floor of the room. (use the formula A = ab.