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kennedy-little

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Modeling with Parametrically Defined Curves
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  1. Modeling with Parametrically Defined Curves Section 10.3

  2. 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.

  3. 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

  4. 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.

  5. 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)

  6. 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

  7. Homework Pages 597 – 598 1 – 6, 8