Honors Math 4 Name:_____________________________ 2D and 3D Parametric Problems 28 March 2018

MORE ⇒

Directions: Each of the following problems involves parametric equations in two or three dimensions. Some of them involve two things moving in different directions at the same time. It might help to draw the situations for the three perspectives of the xy-plane, the yz-plane, and the xz-plane. 1. A stuntman jumps off a building. The equation describing his motion is

€

< 3, 12

t , − 4.9t 2 + t + 6 > . a) Where should the crash mat be placed for him to land safely? b) He practices the jump so he will be ready to jump off the building into the back of a truck full of

trash. The equation for the truck’s motion is

€

< 3t , 4.5 - 4t, 2 + 0.2t > . Will he land in the back of the truck? If so, at what time t?

2. A baseball is hit into left field. Set up the coordinates so that home plate is the origin, first base is on

the positive y-axis and third base is on the negative x-axis. The equation of the baseball’s motion is < −49t, − 4.4t2 +35t, −16t2 +100t + 4 > . The left fielder runs from (-180, 45, 0) parallel to the third base line at a speed of 20 feet per second. Will he catch the ball? If so, at what time t? (Hint: Don’t forget that the player’s arms can extend out from the position of his body.)

3. a) You are given the equation

€

v(t) = cos(t), sin(t), t i. What is the shape of the curve? Can you think of anything natural that looks like this? ii. Make it go twice as fast.

b) You are given the equation

€

v(t) = t2 cos(t), t

2 sin(t), t i. What is the shape of the curve? Can you think of anything natural that looks like this? ii. What is the equation of motion that is 2 seconds behind this one?

Problems 4 through 7 involve Ferris Wheels. For all of them, you may use the plane of the Ferris wheel as the xy-plane. 4. Alex is on a Ferris wheel of radius 35 ft that turns counterclockwise at

the rate of one revolution every 12 seconds. The lowest point of the Ferris wheel (6 o’clock) is 15 feet above ground level at the point (0, 15) on a rectangular coordinate system. Find parametric equations for Alex’s position as a function of time t (in seconds) if the Ferris wheel starts (t = 0) with Alex at (35, 50).

5. A 71-ft radius Ferris wheel turns clockwise one revolution every 20 seconds.

Assume the center of the wheel is at (0, 71). Max stands at a point 90 ft to the left of the base of the wheel (stay in the plane of the wheel). At the instant Ben is at the 9 o’clock position, Max throws a ball from a point 5 feet above the ground toward the Ferris wheel with an initial velocity of 88 ft/sec at an angle with the horizontal of 80°. Find the minimum distance between the ball and Ben. Could Ben catch the ball?

Parametric Application Problems page 2 6. Olivia and Emma are on a ride that is like two Ferris wheels

turning towards each other. Olivia is on a wheel of center (0, 20) and radius 20 ft turning clockwise at the rate of one revolution every 12 seconds. Emma is on another wheel of center (15, 15) and radius 15 ft turning counterclockwise at the rate of one revolution ever 8 seconds. a) Find the minimum distance between Olivia and Emma if both

start out (t = 0) at the 3 o’clock position. b) Find the minimum distance between Olivia and Emma if

Olivia starts out at 3 o’clock and Emma is at the 6 o’clock position.

7. Carlo and Russell are on two Ferris wheels that are next to each other. Carlo is on a Ferris wheel of

center (0, 20, 0) and radius 20 ft turning counterclockwise at the rate of one revolution every 12 seconds. Russell is on another Ferris wheel of center (15, 15, 25) and radius 15 ft turning counterclockwise at the rate of one revolution every 8 seconds. If both boys start at the 3 o’clock position, what is the closest and farthest they get from each other?

8. Parametric equations can be used to describe extremely complicated motions. For example,

compare these two pairs of parametric equations.

x1 = 20costy1 = 20sin t

and x2 = 20cost + 4cos 4 5t( )y2 = 20sin t + 4sin 4 8t( )

Compare and contrast the two curves defined by these parametric equations. 9. A fuel-efficient way to travel from Earth to Mars is to

follow a semi-elliptical orbit known as a Hohmann transfer orbit1. The spacecraft leaves the Earth at the point on the ellipse closest to the sun, and arrives at Mars at the point on the ellipse farthest from the sun, as shown in the figure at right. For the purposes of this problem, assume the orbit of the Earth (eccentricity = 0.0167) and the orbit of Mars (eccentricity = 0.0934) are circular. Let re be the radius of the earth’s orbit, and rm be the radius of Mars’ orbit, and let 2a (=2rmajor) and 2b (=2rminor) be the respective lengths of the horizontal and vertical axes of the ellipse. a) Orienting the ellipse as shown in the figure, with the sun at the origin, find a formula for this

orbit in terms of re, rm, and b. Can you write it in standard and parametric forms? b) It can be shown that b2 = 2are − re

2 . Given this and your answer to part a, find a formula for b in terms of re and rm.

1 Scientific American, March 2000. Note that in an actual Hohmann transfer, the spacecraft would begin in low earth orbit, not from the earth’s surface. Timing is critical in order for the two planets to be correctly aligned. Calculations show that at launch, Mars must lead earth by about 45 degrees, which happens only once every 26 months.

Parametric Application Problems page 3 10. The LORAN (Long-distance Radio Navigation) system is a hyperbolic system. It is used by ships

to navigate at sea2. The ship has a receiver and there is a system of transmitters at set locations. Each trio of transmitters is referred to as a chain. It is a hyperbolic system because the ship determines the difference between its location and that of two transmitters – the geometric definition of a hyperbola. In the diagrams below, A, B, and C are the transmitters, A and B are the foci for the solid line hyperbola and B and C are the foci for the dashed line hyperbola. The ship knows it is at the intersection of the two hyperbolas so it is able to find its location in relation to the map of the transmitters.

Suppose that you are at position P in the figure below and that the three transmitters lie along the vertical axis as shown.

a) Find the standard and parametric equations of a

hyperbola that passes through P and has foci A and B. b) Find the standard and parametric equations of a

hyperbola that passes through P and has foci B and C. c) Sketch the graph of the branches of the hyperbolas

that go through P. d) Find the location of point P.

ANSWER KEY 1. a) position of stuntman at t = 1.2133 is (3, 0.6066, 0)

b) at t = 1, the man is at (3, 0.5, 2.1) and the truck is at (3, 0.5, 2.2) so he lands in the truck and sinks a little into the bags of trash.

2. at t = 6.25 the ball is at (–306.25, 46.875, 4) and the player is at (–305, 45, 0) so he should be able to reach the ball by stretching out his arm.

3. a) i. it is shaped like a spring – vines often climb by growing this way ii. cos(2t), sin(2t), 2t

b) i. shaped like a funnel, tornedos look like this ii. t − 22

cos(t − 2), t − 22

sin(t − 2), t − 2

2 Use has currently been diminished by the use of GPS but still is in use as a backup system.

Parametric Application Problems page 4

4. xA = 35cos2π12

t⎛

⎝⎜

⎞

⎠⎟ , yA = 35sin

2π12

t⎛

⎝⎜

⎞

⎠⎟+ 50

5. xM = 88cos 80°( ) t − 90 , yM = −16t2 +88sin 80°( ) t + 5 and xB = 71cos

2π20

t⎛

⎝⎜

⎞

⎠⎟ , yB = 71sin

2π20

t⎛

⎝⎜

⎞

⎠⎟+ 71

The minimum distance of 1.617 occurs at 2.536 so Ben might not be able to catch the ball.

6. xO = 20cos −2π12

t⎛

⎝⎜

⎞

⎠⎟ , yO = 20sin −

π6t

⎛

⎝⎜

⎞

⎠⎟+ 20

a) xE =15cos2π8t

⎛

⎝⎜

⎞

⎠⎟+15 , yO =15sin

π4t

⎛

⎝⎜

⎞

⎠⎟+15 so min distance of 9.897 occurs at t = 13.993

b) xE =15sinπ4t

⎛

⎝⎜

⎞

⎠⎟+15 , yO = −15cos

π4t

⎛

⎝⎜

⎞

⎠⎟+15 so min distance of 9.98 occurs at t = 55.398

7. xC = 20cosπ6t

⎛

⎝⎜

⎞

⎠⎟ , yC = 20sin

π6t

⎛

⎝⎜

⎞

⎠⎟+ 20 , zC = 0

xR =15cosπ4t

⎛

⎝⎜

⎞

⎠⎟+15 , yR =15sin

π4t

⎛

⎝⎜

⎞

⎠⎟+15 , zR = 25 so min distance of 26.369 occurs at t = 94.28

8. x1, y1 describes a circle of radius 20. x2, y2 describes a “random” path that follows the circle

9. 2a = re + rm so a = re + rm2

and c = a− re =rm − re2

a) x − c( )2

a2+y2

b2=1 replacing c and a with the above and simplifying gives

2x − rm + rc( )2

rm + re( )2+y2

b2=1

b) can start with given relation or a2 − c2 = b2 to get b = rerm

10. a) 2rtrans = 5− 13 and c = 1 − x2

0.514+y− 4( )2

0.486=1 xAB = 0.717tan t yAB =

5− 132

sec t + 4

b) 2rtrans = 13 − 10 and c = 1.5 − x2

2.2+y−1.5( )2

0.049=1 xAB =1.4835tan t yAB =

13 − 102

sec t +1.5

c) see the graph at the right d) The ship could be at (2.06, 1.88) or at (3, 1)