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Chapter 10: Q. 12 (page 667)
In a hyperbola, if and , then ________
In a hyperbola, of and then .
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Uniform Motion A Toyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the
intersection when the Impala is 4 miles from the intersection. See the figure.
(a) Find parametric equations that model the motion of the Camry and Impala.
(b) Find a formula for the distance between the cars as a function of time.
(c) Graph the function in part (b) using a graphing utility.
(d) What is the minimum distance between the cars? When are the cars closest?
(e) Simulate the motion of the cars by simultaneously graphing the equations found in part (a).
Searchlight - A searchlight is shaped like a paraboloid of
revolution. If the light source is located 2 feet from the base
along the axis of symmetry and the depth of the searchlight
is 4 feet, what should the width of the opening be?
Hyperbolic Mirrors Hyperbolas have interesting reflective properties that make them useful for lenses and mirrors. For example, if a ray of light strikes a convex hyperbolic mirror on a line that would (theoretically) pass through its rear focus, it is reflected through the front focus. This property, and that of the parabola, were used to develop the Cassegrain telescope in 1672. The focus of the parabolic mirror and the rear focus of the hyperbolic mirror are the same point. The rays are collected by the parabolic mirror, reflected toward the (common) focus, and thus are reflected by the hyperbolic mirror through the opening to its front focus, where the eyepiece is located. If the equation of the hyperbola is and the focal length (distance from the vertex to the focus) of the parabola is , find the equation of the parabola.
Find an equation for each ellipse. Graph the equation by hand.
Vertices at and : focus at.
The Green MonsterThe left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of 45掳. Use as the acceleration due to gravity and ignore any air resistance.
(a) Find parametric equations that model the position of the ball as a function of time.
(b) What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet.
(c) How far is the ball from home plate at its maximum height? Give your answer in feet.
(d) If the ball is hit straight down the left field wall, will it clear the Green Monster? If it does, by how much does it clear the wall?
Source: The Boston Red Sox
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