Warm Up Plane A is spotted on a rectangular tracking screen, by an air traffic controller at (420,0) flying west 2 miles and north 3.2 miles each minute. 1. Write a linear equation to represent the path of the plane. 2. What is the speed of the plane? (Be sure to include units) 3. Write the parametric equations that model the location of the plane, t minutes after it has entered the screen. Plane B enters the screen and its path is given by the parametric equations x = t and y = t. The planes are flying at the same altitude. 4. Find the point at which the paths of plane A and plane B intersect. 5. Find the amount of time it takes Plane A and Plane B to get to the point of intersection. Do the planes crash?

Homework Answers 1)(a) (150, 75) (b) The crack takes 25 seconds and Rhena takes 20 seconds to get to the point where their paths cross so Rhena gets across the field. 2) x = t, y = – 15t 3) (a) Ranger: y = -6/5x + 120, Lion: y = 1/3x + 51 (b) (45, 66) (c) Ranger’s speed = miles/min. Lion’s speed = miles/min. (d) Ranger: x = 100 – 0.5t, y = t Lion: x = t, y = t (e) Ranger: 110 minutes, Lion: 150 minutes (f) the Ranger leaves the preserve 1 st

Sketching a graph Sketch the curve represented by the parametric equations (indicate the direction using arrows) for -3 < t < 3. Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.

Creating a parametric equation x = y 2 + 3y

Graph the inverse of the function whose parametric equation is x(t) = 2t – 6, y(t) = t 2 – t Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.

Parametric Equations Investigation

Along the River Thames in London, England, there is a huge Ferris wheel known as The British Airways London Eye, but often called the Millennium Ferris Wheel. The Wheel is 135 m (just under 450 ft) high and makes one complete revolution every 15 minutes, in a counterclockwise direction, when the ride is underway. For this exploration, we will reduce the immense passenger capsules to mere points on the rim of a circular wheel. The capsules must clear the ground in their rotation, but we will assume that the radius of the Wheel is the full 67.5 m — exactly one-half the height of the Wheel. This will allow our model to reflect ground level access to the capsule for its passengers.

Recall the radius of the London Eye Ferris wheel was 67.5 meters. Write parametric equations to represent a circle with radius 67.5 meters. Also recall that the wheel revolves once every 15 minutes. How far will the wheel rotate in 1 minute? This is the angular velocity and is the coefficient of t in our parametric equations.

3. Write parametric equations to represent the location of a capsule as a function of time t minutes. Assume that the capsule starts at ground level and the center of the wheel is located at the point (0, 0).

Let the point (0, 0) correspond to the location of a capsule when it is at ground level and let t = 0 represent the time when a particular capsule is at ground level. Write parametric equations that represent the motion of this capsule on the London Eye over the course of a 15 minute ride. Graph the equations on your calculator in an appropriate viewing window. What are the coordinates of the capsule at the beginning and end of the ride? How high off the ground is the capsule after 10 minutes?