Parametric Notes So far we have studied the graphs of several families of functions. Now we are going to look at parametric equations and investigate motion of objects that can be modeled with parametric equations. First, though, we need to look at how we derive parametric equations from familiar functions on the coordinate plane. 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. Parametric Conics The use of two of the three Pythagorean Trigonometric Identities allow for easy parametric representation of ellipses, hyperbolas and circles. Pythagorean Identities

2 2cos sin 1

2 2sec tan 1 (A form of 2 21 tan sec )

For parametric mode, t will be used in place of . t will now represent the angle.

Thus, 2 2cos sin 1 becomes

2 2cos sin 1t t and

2 2sec tan 1 becomes

2 2sec tan 1t t

Circles: Compare the standard form of a circle with the 1st Pythagorean Identity (or the unit circle):

Standard form: 2 2 2x h y k r

Change the equation so that it equals one:

2 2

2 21

x h y k

r r

Pythagorean Identity: 2 2cos sin 1

Using two simple substitutions:

2

2

2cos

x ht

r

and

2

2

2sin

y kt

r

and solving the two equations for ‘x’ and ‘y’ yields a pair of parametric equations: Example:

Graph: 2 2

4 1 16x y

4h 1k 4r

So, 4cos 4tx t and 4sin 1ty t

Try: Rewrite the following equation in parametric form and then graph on your

calculator: 2 2

3 5 9x y

To graph, set calculator: Mode: Parametric

Window: [0, 2]

36 , [-15,15] 1, [-10,10] 1

cosx r t h and siny r t k

Ellipses Horizontal Major Axis Compare the standard form of an ellipse with the 1st Pythagorean Identity:

Pythagorean Identity: 2 2cos sin 1t t

Using the same two simple substitutions:

2

2

2cos

x ht

a

2

2

2sin

y kt

b

And solving the two equations for ‘x’ and ‘y’ yields a pair of parametric equations: Example:

Graph:

2 23 1

116 4

x y

3h 4a 1k 2b

Thus, 4cos 3tx t and 2sin 1ty t

To graph, set calculator: Mode: Parametric

Window: [0, 2] 36 , [-15,15] 1, [-10,10] 1

The endpoints of the major and minor axes can be shown by tracing to the quadrantal angles. Vertical major axis

Example: Graph:

2 22 1

14 9

x y

Thus, 2cos 2tx t and 2sin 1ty t

cosx a t h siny b t k

costx b t h and sinty a t k

Hyperbolas: Horizontal Transverse Axis By using the standard form of a hyperbola with the 2nd Pythagorean Identity:

Standard Form:

2 2

2 21

x h y k

a b

Pythagorean Identity: 2 2sec tan 1

Using two simple substitutions:

2 2

2 2

2 2sec and tan

x h y kt t

a b

and solving the

two equations for ‘x’ and ‘y’ yields a pair of parametric equations: Example: Graph:

2 2

t

3 11

16 4

3 4 1 2

4sec 3 y 2tan 1t

x y

h a k b

x t t

Vertical Transverse Axis Example: Graph:

sec tanx a t h y b t k

1 1tan y secx b t h a t k

2 2

2 11

4 9

1 2 2 3

3 tan 1 and 2sec 2t t

y x

h a k b

x t y t

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

1 sin :x vertical component

cos x : horizontal component

As a result, Horizontal motion can be defined as: 0 cosx v t

Initial velocity time

Vertical motion can be defined as: 2

0

1sin

2y v t gt h

Initial velocity initial height time gravity

Note the value of gravity: feet 1

162

g

meters 1

4.92

g

Parametric Conics Applications

1. The complete graph of the parametric equations 2cosx t and 2siny t 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, 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 0t , 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), 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 40 from 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!

Parametric Applications A

1. A rock is dropped from a 420-foot tower. The rock’s height y in feet above the ground t

seconds later is modeled by 2420 16y t . Find the rock’s position after 1, 1.5, 2.3, 3, and

4.5 seconds. How many second’s does it take before the rock hits the ground? 2. A relief agency drops food containers from an airplane on a war-torn famine area. The drop

was made from an altitude of 1000 feet above ground level.

a. Write parametric equations to model the free fall of the containers b. After 4 seconds of free fall, parachutes open. How many feet above the ground are

the food containers when the parachutes open?

3. Ben can sprint at a rate of 24 feet per second. Jerry sprints at 20 feet per second. Ben

gives Jerry a 10-foot head start. The following parametric equations can be used to model the race.

1 20x t 1 3y

2 24 10x t 2 5y

a. Find a viewing window to simulate a 100-yard dash. Graph simultaneously with t

starting at 0 and Tstep = 0.05. b. Who is ahead after 3 seconds and by how much?

.

4. Nancy hits golf balls off the practice tee with an initial velocity of 180 ft/sec with four different clubs. How far down the fairway does the ball hit the ground if it comes off the club making the specified angle with the horizontal?

a. 15 b. 20 c. 25 d. 30

5. Suppose an ant crawled across the x-y plane in such a way that its position A(x,y) at time t

seconds is given by the two equations: 2

3x t

y t

What is the rectangular equation that represents the path of the ant? Describe the path of the ant in mathematical terms. 6. Consider a spider crawling along the same x-y plane in such a way that it’s position S(x,y) is

given by the equations : 4

6

x t

y t

What is the rectangular equation that represents the path of the spider? Describe the path of the spider in mathematical terms 7. Graph the path of the ant and the spider in parametric mode. Do their paths intersect?

More importantly, at least from the ant’s point of view, will the spider and the ant ever be at the same place at the same point in time?

At what time was the spider at their point of intersection? By how many seconds is the ant spared? 8. Another spider and ant are on a wall. The ant is going in circles that have a radius of 7 cm.

If the wall were mapped out in a coordinate grid, the ant when first observed would be at the point (5,7) and directly left of center of his circular path. The spider is moving on an elliptical path that is 14 cm wide (horizontally) and 6 cm long (vertically). The spider is first observed at (13, 7) and is at the left endpoint of the major axis of his elliptical path.

What rectangular equations represent their paths? What parametric equations represent their paths? Do their paths cross? Are they ever at the same place at the same time? If the ant is moving 7 times as fast as the spider, will they ever be at the same place at the same time? If they are, when will the spider eat the ant? If not, how close do they come to crossing paths and how much time has elapsed since they were first observed?

More Parametric Applications 1) A baseball leaves the baseball player’s bat 3 feet above the ground with an initial velocity of 130 ft/sec at an angle of elevation of 30. If the wind is still, the bat will clear a 20-foot fence 400 feet from home plate. What is the maximum headwind the ball could be hit into and still clear the fence? 2) Another player in the same game hits the ball in the same direction and the same angle of elevation (30) but the ball leaves the bat with a velocity of 120 ft/sec. What tailwind would be needed to get the ball out of the park? 3) Kirby hits a ball when it is 4 ft above the ground with an initial velocity of 120 ft/sec. The ball leaves the bat at a 30 angle with the horizontal and heads toward a 30 ft fence 350 ft from home plate. Suppose that the moment Kirby hits the ball there is a 5 ft/sec wind gust heading directly toward Kirby. Does the ball clear the fence? If so, by how much? If not, could the ball be caught? 4) A dart is thrown upward with an initial velocity of 58 ft/sec at an angle of elevation of 41. Find the parametric equations that model the problem situation. When will the dart hit the ground? Find the maximum height of the dart. When will this occur?

5) A golfer hits a ball with an initial velocity of 133 ft/sec at an angle of elevation of 36. When does the ball hit the ground? Where does the ball hit the ground? Will the ball clear a 28-foot tree that is 275 feet from the golfer? Explain. 6. Sheryl shoots a jump shot from 20 feet straight out from the basket. The ball leaves her hand 8 feet above the floor with an initial velocity of 28 ft/sec and has a takeoff angle of 60. Assume that the front rim extends one foot in front of the backboard and that a shot arriving within a foot behind the backboard (at the 10 foot level) banks in for a score. Did Sheryl make the basket? Explain. 7. Suppose that Tiger hit a golf ball with an initial velocity of 150 ft/sec at an angle of elevation of 30. Write the parametric equations to represent the flight of the ball. How long is the golf ball in the air? Determine the distance the ball traveled. When is the ball at its maximum height? What is the maximum height? 8) Suppose that you are sitting in a chair on a Ferris wheel that has a diameter of 120 feet that is boarded 6 feet above the ground at the 6 o’clock position. If the Ferris wheel moves in a counterclockwise direction and makes one revolution every 12 minutes, how high will your chair be off the ground after 6 minutes? After 32 minutes? At 9 minutes, is your chair going up or going down?

9) A certain amusement park had a giant double Ferris wheel. The double Ferris wheel has a 30-meter rotating arm attached at its center to a 26-meter main support. The arm rotates in the counterclockwise direction, completing one full revolution in 3 minutes. At each end of the rotating arm is attached a Ferris wheel measuring 20 meters in diameter. Each Ferris wheel rotates counterclockwise, completing one full revolution in 2 minutes. At noon, the arm is horizontal and Mary and Chris are in a seat 25 meters from the center and ascending. When will they reach their lowest point and how high will they be then? 10) While waiting in line to ride the Ferris wheel at the state fair, you decide to time the wheel and observe that the time for one complete revolution is 15 seconds. You assume that the diameter of the wheel is 100 feet because a sign beside the wheel says, “Climb 100 feet in the air!” When you get in a seat you note that your seat is 3 feet off the ground. If you ride the Ferris wheel for 190 seconds, how many times and at what times will you be at the top of the wheel? 11) Al and Betty are on a Ferris wheel. The wheel has a radius of 15 feet and its center is 20 feet above the ground. How high are Al and Betty at the 3 o’clock position? At the 12 o’clock position? At the 9 o’clock position? 12) A Ferris wheel has a radius of 20 feet. The center of the wheel is 25 feet above the ground. The wheel makes one revolution every 90 seconds. How long does it take for a person to go from the bottom of the wheel to a position that is 40 feet above the ground?

13) Suppose that a Ferris wheel has a radius of 32 meters and is rotating at 3.5 rpm. Find the

radian measure of at the end of t minutes. Find an expression for the position of the shadow on

the x-axis at the end of t minutes. How long does it take for the position of the shadow on the x-

axis to reach 32x ?

14) A Ferris wheel is 30 meters in diameter and must be boarded from a platform 2 meters above the ground. The wheel completes one full revolution every 12 minutes. What is your height after riding for 112 minutes? 15) A Ferris wheel is 40 feet in diameter and its axle is fixed 32 feet above the ground. Mary got on at time 0t seconds into the car at the bottom of the wheel. She soon discovered that it took 48 seconds to go from the bottom to the top of the Ferris wheel. How high above the ground is Mary after 130 seconds? After 4 minutes?

Pre-Calculus Name______________________________

Parametric Applications Quiz Period_____Date_____________________

1. The center-field fence in a ballpark is 10 feet high and 400 feet from home plate. The baseball is hit 3 feet above the ground. It leaves the bat at an angle with the horizontal at a speed of

120 miles per hour. a) Write a set of parametric equations for the path of the baseball.

______________________________ ______________________________

b) Suppose the baseball is hit at an angle of 23 . Is the hit a home run? _____________

c) If not, find the minimum angle required for the hit to be a homerun. _____________

What window gives you the best picture of the problem situation?

_____________________________________________________ 2. A distress flare is shot straight up from a ship’s bridge 75 ft above the water with an initial velocity of 76 ft/sec. a) Write a set of parametric equations for the path of the flare.

______________________________ ______________________________

What window gives you the best picture of the problem situation?

_____________________________________________________

b) What is the height of the flare after 2 seconds? __________________________ 3. Two opposing players in “Capture the Flag” are 100 ft apart. On a signal, they run to capture a flag that is on the ground midway between them. The faster runner hesitates for 0.1 sec. The following parametric equations model the race to the flag:

1 1

2 2

Runner #1: 10( 0.1), 3

Runner #2: 100 9 , 3

t t

t t

x t y

x t y

Who captures the flag and by how many feet? ______________________ ____________ 4. Tony is launching a yard dart 20 feet from the edge of a circular target of radius 18 in. If Tony throws the dart directly at the target and releases it 3 feet above the ground with an initial velocity of 32 ft/sec at a 70 angle. Hint: Change calculator to radians and degrees in problem to radians. To graph the target, write a set of parametric equations for a circle that ha shifted left 21.5 feet and has a radius of 18 inches.

a) Write a set of parametric equations for the path of the dart.

______________________________ ______________________________

a) Write a set of parametric equations for the target.

______________________________ ______________________________

What window gives you the best picture of the problem situation?

_____________________________________________________

b) Will the dart hit the target?__________

Pre-Calculus H/GT Name______________________________

Parametric Applications Quiz pg 2 Period_____Date_____________________

5. Ron is on a Ferris wheel of radius 40 ft that turns counterclockwise at the rate of one revolution every 15 sec. The lowest point of the Ferris wheel is 10 feet above the ground. Ron is one of the last people to get on the ride. When the ride actually begins he is in a chair whose arm is parallel to the ground.

a) Write a set of parametric equations for Ron’s position as a function of time.

______________________________ ______________________________ b) What is Ron’s exact position 40 seconds after the ride begins?

________________________________________________________________ 6. Bryan hits a baseball when it is 4 feet above the ground with an initial velocity of 120 ft/sec. The ball leaves the bat at a 40 angle with the horizontal and heads toward a 30 ft fence 350 from home plate.

a) Write a set of parametric equations that model the path of the ball.

______________________________ ______________________________

a) Write a set of parametric equations to create the fence.

______________________________ ______________________________

c) Graph both sets of equations in simultaneous mode. Does the ball clear the fence?

______________________________

d) If so, by how much does it clear the fence? If not, could the ball be caught?

_________________________________________________________________

7. A 72 ft radius Ferris Wheel turns counterclockwise one revolution every 16 seconds. Tony stands at a point 90 feet to the right of the base of the wheel. At the instant that Max’s position is parallel to the ground on the side closest to Tony, the ride begins and Tony tosses a ball toward the Ferris wheel with an initial velocity of 88 ft/sec at an angle fo 110 . Find the minimum distance between the ball and Max. Describe/show the steps needed to determine the distance to the nearest thousandth of a foot.

Steps for using List to Calculate Distance **Make sure that the calculator is in parametric mode! Stat, Edit, Enter,up This takes you to List 1 and the command line for the list. 2nd , List, Ops,Seq( This inserts the sequence command.

(expression,variable,begin,end,increment) T,T,0,5,.1), Enter This tells the calculator to determine values from 0 to 5 in

increments of .1 L2,vars,Yvars,parametric,X1t(L1) This pastes X1t in the command line and tells the

calculator to evaluate the equation for all values in L1.

L3,vars,Yvars,parametric,Y1t(L1) This pastes Y1t in the command line and tells the calculator to evaluate the equation for all values in L1.

L4,vars,Yvars,parametric,X2t(L1) This pastes X2t in the command line and tells the calculator to evaluate the equation for all values in L1.

L5 ,vars,Yvars,parametric,Y2t(L1) This pastes Y2t in the command line and tells the calculator to evaluate the equation for all values in L1.

L6 , 2 2

4 2 5 3L L L L This pastes the distance formula for each defined

increment into L6.

OR In Function Mode, you can type the following equation in Y= and find the minimum point (minimum

distance between two points) : 2 2

1 1 2 1 2t t t ty x x x x y x y x