Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Modeling with Parametrically Defined Curves Section 10.3
Example 1 • The Colgate Clock in Jersey City is the world’s largest free-standing clock. It is a circle 50 feet in diameter and the bottom of its face is about 14 feet above the ground. Consider a coordinate system on the ground under the center of the clock. Write the parametric equations fo rhte location of a minute hand after t minutes when t =0 is midnight.
Example 1 • Key info: • Clock – so clockwise! • Starts at the top of the graph • Diameter is 50 feet • Vertical displacement: 14 feet So x(t) = sint y(t) = cos t Amp 50/2 = 25 Start: 39 feet Period : 2π/60
Example 2 • Suppose a planet is orbiting a star in a circular orbit 200 million miles from the center of the star. Suppose a moon is circling the planet 30 times during the orbit of the planet. Give possible parametric equations for the moon.
Example 2 • First do the planet to the star • Assume counterclockwise x(t) = 200cos(t) Y(t) = 200 sin(t) Now, moon to planet x(t) = 1.5cos(30t) y(t) = 1.5 sin(30t) x(t) = 200 cos(t) + 1.5cos(30t) y(t) = 200sin(t) + 1.5sin(30t)
Example 3 • Find a set of parametric equations to model the movement in two dimensions of an 18-inch long pendulum, which swings through a 45˚ arc in about 1.4 seconds. Draw a picture! Period: 2π/2.8 Split into 2 angles: 22.5 18*Sin 22.5 = 6.8883 amp
Homework Pages 597 – 598 1 – 6, 8