T3 2004New Orleans, La

Definition: The graph of ordered pairs (x,y) where the functions are defined on an interval I of t-values is a parametric curve. The equations are parametric equations for the curve and the variable t is a parameter. A parametrization of a curve consists of the parametric equations and the interval of t-values. Time is often the parameter in a problem situation which is why we normally use t for the parameter. Sometimes parametric equations are used by companies in their design plans. It is easier for the company to make larger and smaller objects efficiently by simply changing the parameter t.

The use of two of the three Pythagorean TrigonometricIdentities allow for easy parametric representation ofellipses, hyperbolas and circles. Pythagorean Identities:   (A form of .) For parametric mode, t will be used in place of . t will now represent the angle.

Thus, becomes .

and becomes .

2 2sin cos 1

2 2sec tan 1 2 21 tan sec

2 2sin cos 1 2 2sin cos 1t t

2 2sec tan 1 2 2sec tan 1t t

2 2 2

2 2

2 2

2 2

2 2

2 22 2

(Standard Form)

1

cos sin 1

Substitute: cos and sin

We can now derive a pair of parametric equations:

cos and sin

x h y k r

x h y k

r r

x h y kt t

r r

x r t h y r t k

Circles

Ellipses (Horizontal Major Axis)

2 2

2 2

2 22 2

Pythagorean Identity: cos sin 1

Substitute: cos and sin

We can now derive the parametric equations:

cos and sin

x h y kt t

a b

x a t h y b t k

Ellipses (Vertical Major Axis)

Using the same substitutions with the vertical equations:

cos and cosx b t h y a t k

Hyperbolas (Horizontal Transverse Axis)

2 2

Using Pythagorean Identity:

sec tan 1

We can derive the parametric equations:

sec and tanx a t h y b t k

Hyperbolas (Vertical Transverse Axis)

2 2

Using the same Pythagorean Identity:

sec tan 1

We can derive the parametric equations:

tan and secx b t h y a t k

Parabolas

The trigonometric functions can also be used with parabolas if we think about the unit circle and the concept of vertical and horizontal components as in vectors.

sin x: vertical component

cos x: horizontal component

Horizontal Motion

0 cosx v t

Initial velocity time

Vertical Motion

210 2siny v t gt h

Initial velocity time gravity initial height

*value of gravity: feet

meters

12

12

16 ft/sec

4.9 m/sec

g

g

Parametric Conics Applications

1. The complete graph of the parametric equations and is the circle of radius 2 centered at the origin. Find an interval of values for t so that the graph is the given portion of the circle. a) the portion in the first quadrant.

b) The portion above the x-axis.

c) The portion to the left of the y-axis.

2cosx t 2siny t

2. A baseball is hit straight up from a height of 5 feet with an initial velocity of 80 ft/sec.a) Write an equation that models the height of the ball as a function of time t.b) How high is the ball after 4 seconds?c) What is the maximum height of the ball? How many seconds does it take to reach its maximum height?

3. Kevin hits a baseball at 3 feet above the ground with an initial airspeed of 150 ft/sec at an angle of 18 with the horizontal. Will the ball clear a 20-foot wall that is 400 feet away?

4. Ron is on a Ferris wheel of radius 35 ft. that turns counterclockwise at the rate of on revolution every 12 seconds. The lowest point of the Ferris wheel is 15 feet above ground level at the point, (0,15) on a rectangular coordinate system. Find parametric equations for the position of Ron as a function of time t (in seconds) if the Ferris wheel starts with Ron at the point (35,50).

5. Jane is riding on a Ferris wheel with a radius of 30 feet. The wheel is turning counterclockwise at the rate of one revolution every 10 seconds. Assume the lowest point on the Ferris wheel is 10 feet above the ground. At time , Jane’s seat is on an imaginary line that is parallel to the ground. Find parametric equations to model Jane’s path and then find Jane’s position 22 seconds

into the ride.

6. Chris and Linda warm up in the outfield by tossing softballs to each other as in the picture below. Find the minimum distance between the two balls and when this distance occurs.

7. A Ferris wheel with a 20 ft radius turns counterclockwise one revolution every 12 seconds. Eric stands at point D, 75 feet from the base of the wheel. At the instant Jane is at a point parallel to the ground (see picture on next slide), Eric throws a ball at the Ferris wheel, releasing it from the same height as the bottom of the wheel. If the ball’s initial speed is 60 ft/sec and it is released as an angle of 120 with the horizontal, does Jane have a chance to catch the ball?

8. A Ferris wheel with a 71-foot radius turns counterclockwise one revolution every 20 seconds. Tony stands at a point 90 feet to the right of the base of the wheel. At the instant Mathew is at a point parallel to the ground, Tony throws a ball toward the Ferris wheel with an initial velocity of 88 ft/sec at an angle of 100 with the horizontal. Find the minimum distance between the ball and Mathew.

9. Chang is on a Ferris wheel with a center at (0,20) and a radius of 20 ft. turning counterclockwise at the rate of one revolution every 12 seconds. Kuan is on a Ferris wheel with a center at (15,15) and a radius of 15 ft. turning counterclockwise at the rate of one revolution every 8 seconds. Find the minimum distance between Chang and Kuan if both start out at a position parallel to the ground (3 o’clock).

10. Chang and Kuan are riding on the Ferris wheel as in the problem above. With only one revolution of the wheels, find the minimum distance between Chang and Kuan if Chang starts out at a point parallel to the ground (3 o’clock) and Kuan starts out at the lowest position possible (6 o’clock).

*11. Jaime and her friends find a ball toss game at the Carson Carnival. Jaime must throw

the basketball from a line marked on the ground18 feet away from the target. Jaime releases the ball from a height of 6 feet with an initial velocity of 30 feet per second and an angle of 40from the horizontal. The ball must go into a square box that is on top of two poles. The top of the box is 10 feet from the ground. If the ball has a diameter of 12 inches and the box is 24 inches wide, will the ball go into the box? Use your work and explain why or why not. If not, how should she change the way she throws the ball?

12. Jenny, who is 5 feet tall, is standing on top of a 40-foot building. A taller building is 25 feet from this building. The taller building is 60 feet tall and 30 feet wide. How might she throw the ball so that it would land on the roof of the taller building?

13. A baseball is hit from a height of 3 feet above the ground. It leaves the bat with an initial velocity of 152 ft/sec at an angle of elevation of 20. A 24 foot fence is located 400 feet away from home plate. If there is an 8 mph wind blowing directly at the batter, will the ball go over the fence? If not, what is the smallest angle at which the ball can leave the bat and be a home run? Use only parametric equations to model the entire problem situation. Explain your answers.

*14. An Air Traffic Controller is monitoring the progress of two planes. When he first makes note of the plane’s positions, Plane A is 400 miles due north of the control center and Plane B is 300 miles east of the control center. A minute later, he notices that Plane A is 6 miles east of and 8 miles south of its previous position, while Plane B is 5 miles west and 7 miles north of its previous position. The planes are currently flying at the same altitude on direct routes. Will the controller need to change the flight path of one of the planes?

15. A wildlife preserve extends 80 miles north and 120 miles east of the ranger station. The ranger leaves from a point 100 miles east of the station along the southern boundary to survey the area. He travels 0.6 miles north and 0.5 miles west every minute. A lion leaves the west edge of the preserve 51 miles north of the station at the same time the ranger leaves his position at the southern boundary. Every minute, the lion moves 0.1 miles north and 0.3 miles east. Do the ranger and the lion collide? Sketch the problem situation, then use both linear and parametric equations to model. Explain in detail what happens and why you believe they do or do not collide. Your answers using one method should validate the answers derived using the other!

A copy of the entire presentation can be found at:

www.fortbend.k12.tx.us/campuses/dhs/courses

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