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Texas Instruments 83CML/ILI/U Guidebook - Page 332

Using Parametric Equations: Ferris Wheel Problem

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Using Parametric Equations: Ferris Wheel Problem Problem Procedure Using two pairs of parametric equations, determine when two objects in motion are closest to each other in the same plane. A ferris wheel has a diameter (d) of 20 meters and is rotating counterclockwise at a rate (s) of one revolution every 12 seconds. The parametric equations below describe the location of a ferris wheel passenger at time T, where a is the angle of rotation, (0,0) is the bottom center of the ferris wheel, and (10,10) is the passenger's location at the rightmost point, when T=0. X(T) = r cos a where a = 2pTs and r = d à 2 Y(T) = r + r sin a A person standing on the ground throws a ball to the ferris wheel passenger. The thrower's arm is at the same height as the bottom of the ferris wheel, but 25 meters (b) to the right of the ferris wheel's lowest point (25,0). The person throws the ball with velocity (v0) of 22 meters per second at an angle (q) of 66¡ from the horizontal. The parametric equations below describe the location of the ball at time T. X(T) = b N Tv0 cosq Y(T) = Tv0 sinq N (g à 2 ) T2 9.8 m / sec2 where g = 1. Press z. Select Par, Simul, and the default settings. Simul (simultaneous) mode simulates the two objects in motion over time. 2. Press p. Set the viewing window. Tmin=0 Xmin=L13 Tmax=12 Xmax=34 Tstep=.1 Xscl=10 Ymin=0 Ymax=31 Yscl=10 3. Press o. Turn off all functions and stat plots. Enter the expressions to define the path of the ferris wheel and the path of the ball. Set the graph style for X2T to ë (path). Tip: Try setting the graph styles to ë X1T and ì X2T, which simulates a chair on the ferris wheel and the ball flying through the air when you press s. 17-12 Applications

Section	Page
TI-83 Graphing Calculator Guidebook	1
Important and US FCC Information	2
Table of Contents	3
Getting Started: Do This First!	3
Chapter 1: Operating the TI- 83	3
Chapter 2: Math, Angle, and Test Operations	4
Chapter 3: Function Graphing	4
Chapter 4: Parametric Graphing	4
Chapter 5: Polar Graphing	4
Chapter 6: Sequence Graphing	5
Chapter 7: Tables	5
Chapter 8: DRAW Operations	5
Chapter 9: Split Screen	5
Chapter 10: Matrices	6
Chapter 11: Lists	6
Chapter 12: Statistics	6
Chapter 13: Inferential Statistics and Distributions	6
Chapter 14: Financial Functions	7
Chapter 15: CATALOG, Strings, Hyperbolic Functions	7
Chapter 16: Programming	7
Chapter 17: Applications	7
Chapter 18: Memory Management	8
Chapter 19: Communication Link	8
Appendix A: Tables and Reference Information	8
Appendix B: General Information	8
Index	8
Getting Started: Do This First!	9
TI-83 Keyboard	10
Keyboard Zones	10
Using the Color- Coded Keyboard	11
Using the [2nd] and [ALPHA] Keys	11
TI-83 Menus	12
Displaying a Menu	12
Moving from One Menu to Another	12
Selecting an Item from a Menu	12
Leaving a Menu without Making a Selection	12
First Steps	13
The Quadratic Formula	14
Entering a Calculation	14
Converting to a Fraction	15
Displaying Complex Results	16
Box with Lid	17
Defining a Function	17
Defining a Table of Values	18
Zooming In on the Table	19
Setting the Viewing Window	20
Displaying and Tracing the Graph	21
Zooming In on the Graph	23
Finding the Calculated Maximum	24
Other TI-83 Features	25
Graphing	25
Sequences	25
Tables	25
Split Screen	25
Matrices	25
Lists	25
Statistics	25
Inferential Statistics	26
Financial Functions	26
CATALOG	26
Programming	26
Communication Link	26
Chapter 1: Operating the TI- 83	27
Turning On and Turning Off the TI-83	28
Turning On the Calculator	28
Turning Off the Calculator	28
Batteries	28
Setting the Display Contrast	29
Adjusting the Display Contrast	29
When to Replace Batteries	29
The Display	30
Types of Displays	30
Home Screen	30
Displaying Entries and Answers	30
Returning to the Home Screen	30
Busy Indicator	30
Display Cursors	31
Entering Expressions and Instructions	32
What Is an Expression?	32
Entering an Expression	32
Multiple Entries on a Line	32
Entering a Number in Scientific Notation	33
Functions	33
Instructions	33
Interrupting a Calculation	33
TI-83 Edit Keys	34
Setting Modes	35
Checking Mode Settings	35
Changing Mode Settings	35
Setting a Mode from a Program	35
Normal, Sci, Eng	36
Float, 0123456789	36
Radian, Degree	37
Func, Par, Pol, Seq	37
Connected, Dot	37
Sequential, Simul	38
Real, a+bi, re^bi	38
Full, Horiz, G.T	38
Using TI-83 Variable Names	39
Variables and Defined Items	39
Notes about Variables	39
Storing Variable Values	40
Storing Values in a Variable	40
Displaying a Variable Value	40
Recalling Variable Values	41
Using Recall (RCL)	41
ENTRY (Last Entry) Storage Area	42
Using ENTRY (Last Entry)	42
Accessing a Previous Entry	42
Reexecuting the Previous Entry	43
Multiple Entry Values on a Line	43
Clearing ENTRY	43
Ans (Last Answer) Storage Area	44
Using Ans in an Expression	44
Continuing an Expression	44
Storing Answers	44
TI-83 Menus	45
Using a TI-83 Menu	45
Scrolling a Menu	45
Selecting an Item from a Menu	46
Leaving a Menu without Making a Selection	46
VARS and VARS Y-VARS Menus	47
VARS Menu	47
Selecting a Variable from the VARS Menu or VARS Y- VARS Menu	47
Equation Operating System (EOS™)	48
Order of Evaluation	48
Implied Multiplication	49
Parentheses	49
Negation	49
Error Conditions	50
Diagnosing an Error	50
Correcting an Error	50
Chapter 2: Math, Angle, and Test Operations	51
Getting Started: Coin Flip	52
Keyboard Math Operations	53
Using Lists with Math Operations	53
Addition, Subtraction, Multiplication, Division	53
Trigonometric Functions	53
Power, Square, Square Root	53
Inverse	53
log(, 10^(, ln(	54
Exponential	54
Constant	54
Negation	54
Pi	54
MATH Operations	55
MATH Menu	55
>Frac, >Dec	55
Cube, Cube Root	56
Root	56
fMin(, fMax(	56
nDeriv(	57
fnInt(	57
Using the Equation Solver	58
Solver	58
Entering an Expression in the Equation Solver	58
Entering and Editing Variable Values	59
Solving for a Variable in the Equation Solver	60
Editing an Equation Stored to eqn	62
Equations with Multiple Roots	62
Further Solutions	62
Controlling the Solution for Solver or solve(	62
Using solve( on the Home Screen or from a Program	62
MATH NUM (Number) Operations	63
MATH NUM Menu	63
abs(	63
round(	63
iPart(, fPart(	64
int(	64
min(, max(	65
lcm(, gcd(	65
Entering and Using Complex Numbers	66
Complex-Number Modes	66
Entering Complex Numbers	66
Note about Radian versus Degree Mode	66
Interpreting Complex Results	67
Rectangular-Complex Mode	67
Polar-Complex Mode	67
MATH CPX (Complex) Operations	68
MATH CPX Menu	68
conj(	68
real(	68
imag(	68
angle(	69
abs(	69
>Rect	69
>Polar	69
MATH PRB (Probability) Operations	70
MATH PRB Menu	70
rand	70
nPr, nCr	71
! (Factorial)	71
randInt(	72
randNorm(	72
randBin(	72
ANGLE Operations	73
ANGLE Menu	73
DMS Entry Notation	73
Degree	73
Radians	74
R>Pr (, R>Pq(, P>Rx(, P>Ry(	74
TEST (Relational) Operations	75
TEST Menu	75
=, ne, >, ge, <, le	75
Using Tests	75
TEST LOGIC (Boolean) Operations	76
TEST LOGIC Menu	76
Boolean Operators	76
and, or, xor	76
not(	76
Using Boolean Operations	76
Chapter 3: Function Graphing	77
Getting Started: Graphing a Circle	78
Defining Graphs	79
TI-83—Graphing Mode Similarities	79
Defining a Graph	79
Displaying and Exploring a Graph	79
Saving a Graph for Later Use	79
Setting the Graph Modes	80
Checking and Changing the Graphing Mode	80
Setting Modes from a Program	80
Defining Functions	81
Displaying Functionsin the Y= Editor	81
Defining or Editing a Function	81
Defining a Function from the Home Screen or a Program	82
Evaluating Y= Functions in Expressions	82
Selecting and Deselecting Functions	83
Selecting and Deselecting a Function	83
Turning On or Turning Off a Stat Plot in the Y= Editor	83
Selecting and Deselecting Functions from the Home Screen or a Program	84
Setting Graph Styles for Functions	85
Graph Style Icons in the Y= Editor	85
Setting the Graph Style	85
Shading Above and Below	86
Setting a Graph Style from a Program	86
Setting the Viewing Window Variables	87
The TI-83 Viewing Window	87
Displaying the Window Variables	87
Changing a Window Variable Value	87
Storing to a Window Variable from the Home Screen or a Program	88
Delta X and Delta Y	88
Setting the Graph Format	89
Displaying the Format Settings	89
Changing a Format Setting	89
RectGC, PolarGC	89
CoordOn, CoordOff	90
GridOff, GridOn	90
AxesOn, AxesOff	90
LabelOff, LabelOn	90
ExprOn, ExprOff	90
Displaying Graphs	91
Displaying a New Graph	91
Pausing or Stopping a Graph	91
Smart Graph	91
Overlaying Functions on a Graph	92
Graphing a Family of Curves	92
Exploring Graphs with the Free-Moving Cursor	93
Free-Moving Cursor	93
Graphing Accuracy	93
Exploring Graphs with TRACE	94
Beginning a Trace	94
Moving the Trace Cursor	94
Moving the Trace Cursor from Function to Function	94
Moving the Trace Cursor to Any Valid X Value	95
Panning to the Left or Right	95
Quick Zoom	95
Leaving and Returning to TRACE	95
Using TRACE in a Program	95
Exploring Graphs with the ZOOM Instructions	96
ZOOM Menu	96
Zoom Cursor	96
ZBox	96
Zoom In, Zoom Out	97
ZDecimal	97
ZSquare	97
ZStandard	98
ZTrig	98
ZInteger	98
ZoomStat	98
ZoomFit	98
Using ZOOM MEMORY	99
ZOOM MEMORY Menu	99
ZPrevious	99
ZoomSto	99
ZoomRcl	99
ZOOM FACTORS	100
Checking XFact and YFact	100
Changing XFact and YFact	100
Using ZOOM MEMORY Menu Items from the Home Screen or a Program	100
Using the CALC (Calculate) Operations	101
CALCULATE Menu	101
value	101
zero	102
minimum, maximum	103
intersect	103
dy/dx	104
Integral f(x)dx	104
Chapter 4: Parametric Graphing	105
Getting Started: Path of a Ball	106
Defining and Displaying Parametric Graphs	108
TI-83 Graphing Mode Similarities	108
Setting Parametric Graphing Mode	108
Displaying the Parametric Y= Editor	108
Selecting a Graph Style	108
Defining and Editing Parametric Equations	109
Selecting and Deselecting Parametric Equations	109
Setting Window Variables	109
Setting the Graph Format	110
Displaying a Graph	110
Window Variables and Y- VARS Menus	110
Exploring Parametric Graphs	111
Free-Moving Cursor	111
TRACE	111
Moving the Trace Cursor to Any Valid T Value	112
Chapter 5: Polar Graphing	113
Getting Started: Polar Rose	114
Defining and Displaying Polar Graphs	115
TI-83 Graphing Mode Similarities	115
Setting Polar Graphing Mode	115
Displaying the Polar Y= Editor	115
Selecting Graph Styles	115
Defining and Editing Polar Equations	116
Selecting and Deselecting Polar Equations	116
Setting Window Variables	116
Setting the Graph Format	117
Displaying a Graph	117
Window Variables and Y-VARS Menus	117
Exploring Polar Graphs	118
Free-Moving Cursor	118
TRACE	118
Moving the Trace Cursor to Any Valid theta Value	118
ZOOM	118
CALC	118
Chapter 6: Sequence Graphing	119
Getting Started: Forest and Trees	120
Defining and Displaying Sequence Graphs	121
TI-83 Graphing Mode Similarities	121
Setting Sequence Graphing Mode	121
TI-83 Sequence Functions u, v, and w	121
Displaying the Sequence Y= Editor	122
Selecting Graph Styles	122
Selecting and Deselecting Sequence Functions	122
Defining and Editing a Sequence Function	123
Nonrecursive Sequences	123
Recursive Sequences	124
Setting Window Variables	125
Selecting Axes Combinations	126
Setting the Graph Format	126
Setting Axes Format	126
Displaying a Sequence Graph	126
Exploring Sequence Graphs	127
Free-Moving Cursor	127
TRACE	127
Moving the Trace Cursor to Any Valid n Value	127
ZOOM	128
CALC	128
Evaluating u, v, and w	128
Graphing Web Plots	129
Graphing a Web Plot	129
Valid Functions for Web Plots	129
Displaying the Graph Screen	129
Drawing the Web	129
Using Web Plots to Illustrate Convergence	130
Example: Convergence	130
Graphing Phase Plots	131
Graphing with uv, vw, and uw	131
Example: Predator-Prey Model	131
Comparing TI-83 and TI-82 Sequence Variables	133
Sequences and Window Variables	133
Keystroke Differences Between TI-83 and TI-82	134
Sequence Keystroke Changes	134
Chapter 7: Tables	135
Getting Started: Roots of a Function	136
Setting Up the Table	137
TABLE SETUP Screen	137
TblStart, Delta Tbl	137
Indpnt: Auto, Indpnt: Ask, Depend: Auto, Depend: Ask	137
Setting Up the Table from the Home Screen or a Program	137
Defining the Dependent Variables	138
Defining Dependent Variables from the Y= Editor	138
Editing Dependent Variables from the Table Editor	138
Displaying the Table	139
The Table	139
Independent and Dependent Variables	139
Clearing the Table from the Home Screen or a Program	139
Scrolling Independent-Variable Values	140
Displaying Other Dependent Variables	140
Chapter 8: Draw Instructions	141
Getting Started: Drawing a Tangent Line	142
Using the DRAW Menu	143
DRAW Menu	143
Before Drawing on a Graph	143
Drawing on a Graph	143
Clearing Drawings	144
Clearing Drawings When a Graph Is Displayed	144
Clearing Drawings from the Home Screen or a Program	144
Drawing Line Segments	145
Drawing a Line Segment Directly on a Graph	145
Drawing a Line Segment from the Home Screen or a Program	145
Drawing Horizontal and Vertical Lines	146
Drawing a Line Directly on a Graph	146
Drawing a Line from the Home Screen or a Program	147
Drawing Tangent Lines	148
Drawing a Tangent Line Directly on a Graph	148
Drawing a Tangent Line from the Home Screen or a Program	148
Drawing Functions and Inverses	149
Drawing a Function	149
Drawing an Inverse of a Function	149
Shading Areas on a Graph	150
Shading a Graph	150
Drawing Circles	151
Drawing a Circle Directly on a Graph	151
Drawing a Circle from the Home Screen or a Program	151
Placing Text on a Graph	152
Placing Text Directly on a Graph	152
Placing Text on a Graph from the Home Screen or a Program	152
Split Screen	152
Using Pen to Draw on a Graph	153
Using Pen to Draw on a Graph	153
Drawing Points on a Graph	154
DRAW POINTS Menu	154
Drawing Points Directly on a Graph with Pt- On(	154
Erasing Points with Pt- Off(	155
Changing Points with Pt- Change(	155
Drawing Points from the Home Screen or a Program	155
Drawing Pixels	156
TI-83 Pixels	156
Turning On and Off Pixels with Pxl- On( and Pxl- Off(	156
Using pxl-Test(	156
Split Screen	156
Storing Graph Pictures (Pics)	157
DRAW STO Menu	157
Storing a Graph Picture	157
Recalling Graph Pictures (Pics)	158
Recalling a Graph Picture	158
Deleting a Graph Picture	158
Storing Graph Databases (GDBs)	159
What Is a Graph Database?	159
Storing a Graph Database	159
Recalling Graph Databases (GDBs)	160
Recalling a Graph Database	160
Deleting a Graph Database	160
Chapter 9: Split Screen	161
Getting Started: Exploring the Unit Circle	162
Using Split Screen	163
Setting a Split-Screen Mode	163
Horiz (Horizontal) Split Screen	164
Horiz Mode	164
Moving from Half to Half in Horiz Mode	164
Full Screens in Horiz Mode	164
G-T (Graph-Table) Split Screen	165
G-T Mode	165
Moving from Half to Half in G- T Mode	165
Using [TRACE] in G-T Mode	165
Full Screens in G.T Mode	165
TI-83 Pixels in Horiz and G-T Modes	166
TI-83 Pixels in Horiz and G- T Modes	166
DRAW POINTS Menu Pixel Instructions	166
DRAW Menu Text( Instruction	166
PRGM I/O Menu Output( Instruction	166
Setting a Split- Screen Mode from the Home Screen or a Program	166
Chapter 10: Matrices	167
Getting Started: Systems of Linear Equations	168
Defining a Matrix	169
What Is a Matrix?	169
Selecting a Matrix	169
Accepting or Changing Matrix Dimensions	169
Viewing and Editing Matrix Elements	170
Displaying Matrix Elements	170
Deleting a Matrix	170
Viewing a Matrix	171
Viewing-Context Keys	171
Editing a Matrix Element	172
Editing-Context Keys	172
Using Matrices with Expressions	173
Using a Matrix in an Expression	173
Entering a Matrix in an Expression	173
Displaying and Copying Matrices	174
Displaying a Matrix	174
Copying One Matrix to Another	174
Accessing a Matrix Element	174
Using Math Functions with Matrices	175
Using Math Functions with Matrices	175
Add, Subtract, Multiply	175
Negation	175
abs(	176
round(	176
Inverse	176
Powers	176
Relational Operations	177
iPart(, fPart(, int(	177
Using the MATRX MATH Operations	178
MATRX MATH Menu	178
det(	178
Transpose	178
Accessing Matrix Dimensions with dim(	178
Creating a Matrix with dim(	179
Redimensioning a Matrix with dim(	179
Fill(	179
identity(	179