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x 2y˛ 2 12y 14. Label all character- Solution Rewrite the equation and complete the square in y, being careful to add the appropriate amounts to both sides of the equation. 2 1 1 2 2 12y x 14 2y˛ x 14 2 6y y˛ 2 2 x 14 2 2 6y 9 y˛ 2 2 x 4 y 22 2 1 1 2 1 1 1 2 22 9 1 2 Thus, the graph is the graph of the parabola y˛ 2 1... |
, k a) (h, k c) and c 2a2 b2 where x h y k • If a and b are positive real numbers, then the graph of each of the following equations is a hyperbola with center (h, k). (x h)2 a2 (y k)2 b2 1 (y k)2 a2 (x h)2 b2 focal axis on the horizontal line (h a, k) vertices: foci: and (h c, k), (h a, k) (h c, k) and c 2a2 b2 asympt... |
degree equation is not in standard form, the fastest way to graph it is to use the method in Example 5, modified as in the next example. Example 6 Graph a Conic Not in Standard Form 2 8y˛ 2 6x 9y 4 0 x˛ without putting it in Graph the equation standard form. Solution Write the equation as This is a quadratic equation o... |
D 0, E 12, the and form of equation F 6. [1] with Conversely, it can be shown that the graph of every second-degree equation is a conic section (possibly degenerate—see page 691). When the equation has an xy term, the conic may be rotated from standard position such that its axis or axes are not parallel to the coordi... |
B 4, C 3. 2 4 2 3 16 24 8 and Hence, the graph is an ellipse, a circle, or a single point. Use the quadratic formula to solve for y. 2 3y 4x 6 y 2x 2 5x 8 0 2 Graph both solutions on the same screen, as shown in Figure 11.4–8. 1 2 1 1 y 4x 6 ± 2 1 2 4x 6 2 4 3 2 2 3 2x 2 5x 8 1 2 ■ 724 Chapter 11 Analytic Geometry Exa... |
later, a signal from Q. It also receives a signal from R 305 microseconds after the one from P. Determine the ship’s location. Solution Let the x-axis be the line through the LORAN stations, with the origin located midway between Q and P, so that the situation looks like Figure 11.4-10. If the signal takes t microseco... |
, you can verify that 0 d1 980 305 298,900 feet 56.61 miles. a 56.61 2 28.305 and a˛ 2 28.305˛ 200 c, k 1 801.17. 100, 0 This hyperbola has center (200, 0) and its foci are and, 300, 0 2 100˛ The ship also lies on the hyperbola which implies that 2 801.17 9198.83 200 c, k 1 2 2 a˛ b2 c˛ 2 1 2 1 c 100. 2 x 200 801.17 1 ... |
2, 3 2 3110, 11 2 1 ; vertex 6. hyperbola with center passing through A 5, 1 ; 2 1 1, 1 413 2, 1 3, 1 ; 2 ; 2 1 1 B vertex B 7. hyperbola with center (4, 2); vertex (7, 2); asymptote 3y 4x 10 8. hyperbola with center asymptote 1 6y 5x 15 3, 5 ; vertex 2 3, 0 ; 2 1 9. parabola with vertex (1, 0); axis x 1; passing thro... |
inant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. 35. 9x2 4y2 54x 8y 49 0 36. 4x˛ 2 5y˛ 2 8x 30y 29 0 Section 11.4 Translations and Rotations of Conics 727 37. 4y˛ 2 x˛ 2 6x 24y 11 0 38. 2 16y˛ x˛ 2 0 39. 3y˛ 2 x 2y 1 0 40. 2 6x y 5 0 x˛ 41. 41x˛ 2 24xy ... |
the parabola. passes through 2 dx 4 y 1 6, 3 1 2 1 2 59. Find the points of intersection of the parabola 4y˛ 2 4y 5x 12 and the line x 9. 60. Find the points of intersection of the parabola 4x˛ 2 8x 2y 5 and the line y 15. In Exercises 61–64, write the resulting equation in standard form. 61. Translate the hyperbola d... |
system with x-axis through P and Q, and origin midway between them. 728 Chapter 11 Analytic Geometry 11.4.A Excursion: Rotation of Axes Objectives The graph of an equation of the form • Write the equation of a rotated conic section in terms of u and v • Determine the angle of rotation of a rotated conic section y u Ax... |
r sin u b r sin b 2 1 2 sin u A similar argument with sine leads to the following result. 1 and the addition identity for 2 The Rotation Equations If the xy coordinate axes are rotated through an angle to produce the uv coordinate axes, then the coordinates (x, y) and (u, v) of a point are related by the following equ... |
preceding example changed the original equation, which included an xy term, to an equation that had no uv term. This can be done for any second-degree equation by choosing an angle of rotation that will eliminate the xy term. 2 Dx Ey F 0 (B 0) Au2 Cv2 Du Ev F 0 by Ax˛ 2 Bxy Cy˛ The equation can be rewritten as rotatin... |
xy term were found in the preceding example. Substitute the rotation equations into the given equation and simplify the result to eliminate the xy term. 732 Chapter 11 Analytic Geometry 153x˛ 153 a 97 153 a 2 2 b a a 3 4 192 1710 25 uv 9 2 24 25 v˛ 25 u2 24 16 25 u˛ 97 9 a a 2 4 2 192xy 97y 1710x 1470y 5625 0 5 u 4 5 ... |
Ey F 0 Ax2 Bxy Cy2 Dx B 0 u and an angle with equations to rewrite the equation in the form A¿u2 B¿uv C¿v2 D¿u E¿v F¿ 0,, use the rotation are expressions involving and the constants A,..., F. where sin u, A¿,..., F¿ cos u b. Verify that B¿ 2 C A cos2 u sin2 u 2 2 1 c. Use the double-angle identities to show that sin ... |
If the rotation from the xy-coordinate system to the uv-coordinate system is positive, then the rotation from the uv-coordinate system to the xy-coordinate system is negative. by the in the 2 x, y u, v 2 1 u x cos u y sin u v y cos u x sin u. In Exercises 13–16, find the new coordinates of the point when the coordinat... |
because 5p 3 7p 3 2, a and, 5p 3 b ˛ are coterminal angles, the coordinates p 3b 2, a, 2, a 7p 3 b, and all represent the same point, as shown in Figure 11.5-3. p 3, π 3) ( 2, ( 2, 7π 3 ) 2, – 5π ( 3 ) π 3 O polar axis O 7π 3 polar axis Figure 11.5-3 O – 5π 3 polar axis the point The r-coordinate may also be negative,... |
as x, y 1 2, as shown y-axis =θ π 2 (r, ) P θ (x, y) r θ polar axis Figure 11.5-6 x-axis = 0 θ Let r be as shown in Figure 11.5-6, with r positive. Since r is the distance the distance formula shows that from (0, 0) to x, y, 1 2 r 2x2 y2 Also, by the definitions of the trigonometric functions in the coordinate plane, ... |
has no solutions when x 0 u 738 Chapter 11 Analytic Geometry u tan 1 y x kp (k is any integer) Not every solution works for a specific point P. To find solutions that represent the point P, you need to know which quadrant contains P. u tan can be used for points in Quadrants I and IV because it would 1 y x be an angle... |
Polar Graphs r 2 cos u, where r and are the variables, is a polar An equation like equation. Equations in x and y are called rectangular or Cartesian equations. Many useful curves have simple polar equations, although they may have complicated rectangular equations. u u Like other graphs, the graph of a polar equation... |
1 + sin increases from 1 to 2. θ π 2 θ θAs increases from to π, sin decreases from 1 to 0. So r = 1 + sin decreases from 2 to 1= π 1 O 1 θ As increases from π to, sin decreases from 0 to −1. So r = 1 + sin decreases from 1 to 0. θ 3π 2 θ θ As increases from to 2π, sin increases from −1 to 0. So r = 1 + sin increases f... |
? Common Polar Graphs The following is a summary of commonly encountered polar graphs. In each case, a and b are constants, and is measured in radians. u Depending on the plus or minus sign and whether sine or cosine is used, the basic shape of each graph may differ from those shown by a rotation, reflection, or horizo... |
the given point, each with a different r 77 0, combination of signs (that is, U 66 0; r 77 0, r 66 0, U 66 0). r 66 0, U 77 0; U 77 0; and 2. 5. p 3 b 3, a 3. 1 5, p 2 4. 2, 2p a 3 b 1, p 6 b a 6. 13, a 3p 4 b π R Q P 1 7π 6 S T 3 5 V 7 polar axis In Exercises 7–10, convert the polar coordinates to rectangular coordin... |
has coordinates and verify that the point Q with rectangular coordinates x, y 2 1 r 6 0, r proved in the text apply to Q. For instance, x r cos u, has polar coordinates is positive and the conversion formulas which implies that 2 r, u x r cos u. r 6 0, Hint: If Since with r, u,. 1 2 1 2 1 51. Critical Thinking Distanc... |
polar equation of a conic section NOTE Do not confuse the eccentricity of a conic section, which is denoted as e and whose value varies, with the number e, which is the constant 2.718281828.... The meaning should be clear in context. In a rectangular coordinate system, each type of conic section has a different defini... |
�2 21 1 Solution a. 2 y˛ 4 x˛2 21 1 b. 4x˛2 9y˛2 32x 90y 253 0 represents a hyperbola with a2 4 and b2 21, so e 2a˛2 b2 a 14 21 5 2 2 4x2 9y2 32x 90y 253 0 125 2 2.5 can b. From Example 2 in Section 11.4, be written in standard form, as The graph of this equation has the same shape as the ellipse 2 x˛ 9 2 y˛ 4 1 with a... |
the polar equations of conic sections. Alternate Definition of Conic Sections Let L be a fixed line called a directrix, P a fixed point not on L, and e a positive constant. The set of all points X in the plane such that distance between X and the fixed point distance between X and the fixed line XP XL e is a conic sec... |
can be shown that Figure 11.6-6 r ed 1 e cos u If L is a horizontal line, it can also be shown that r ed 1 e sin u or r ed 1 e sin u depending on whether L is below the pole or above it. If an equation has another value in place of 1, divide both numerator and denominator by that number to rewrite the equation in the ... |
0.7 cos u 2 2.1 1 0.7 cos u r 1 3 1 1 1 cos u 2 3 1 cos u r 3 2 1 1 2 cos u 2 6 1 2 cos u ellipse 10 parabola 10 15 15 15 15 −15 10 10 Figure 11.6-7 hyperbola 10 −10 15 ■ Example 3 Polar Equations of Conic Sections Identify the conic section that is the graph of r 20 4 10 sin u and find its eccentricity and vertices. ... |
value of e into the equation for d shows that and solving and the equation of the ellipse is d 12. ed 4 Hence, 3 1 2 1 e ed r 4 1 1 3 cos u or equivalently, r 12 3 cos u If you had started this process with the equation r ed 1 e cos u, you would have obtained is always positive. e 1 3, which is impossible since the ec... |
2 x˛2 14 16 y˛2 6 b. Compute the eccentricity of each ellipse in part a. c. Based on parts a and b, how is the shape of an 1 x˛2 16 1 x˛2 16 y˛2 1 1 ellipse related to its eccentricity? 20. a. Graph these hyperbolas on the same screen, if possible. y˛2 x˛2 1 4 1 y˛2 4 x˛2 12 1 y˛2 4 x˛2 96 1 b. Compute the eccentricity... |
sun as the pole 754 Chapter 11 Analytic Geometry and assuming the axis of the orbit lies along the polar axis, find a polar equation for the orbit. 48. Halley’s Comet has an elliptical orbit, with eccentricity 0.97 and the sun as a focus. The length of the major axis of the orbit is 3364.74 million miles. Using the su... |
is called a parameterization of the curve. More than one parameterization is possible for a given curve. Example 1 Parameterizations of a Line Find three parameterizations of the line through 1, 3 1 2 with slope 2. Solution The equation of the line in rectangular coordinates is y 3 2 x 1 1 2 or equivalently y 2x 1 [1]... |
t. Find several points by picking values for t, finding the corresponding values of x and y, plotting the points, and connecting the points in the order determined by the least to greatest values of t. t 1 0 1 2 x 2t y 4t˛ 2 4 0 4 0 12 2 0 2 4 y 12 8 4 (x, y) 2, 0 2 1 0, 4 1 2 2, 0 2 1 4, 12 2 1 14 −8 0 −4 −4 x 4 8 10... |
eterization of Transformations Given the parent relation represent the relation, and sketch the graph. x y˛ 2, write a set of parametric equations to Then write the parametric equations of the following successive transformations of the parent relation, and sketch each graph. 758 Chapter 11 Analytic Geometry y 4 2 0 −2... |
Parabola A golfer hits a ball with an initial velocity of 140 feet per second so that with the horizontal. its path as it leaves the ground makes an angle of 31° a. When does the ball hit the ground? b. How far from its starting point does it land? c. What is the maximum height of the ball during its flight? NOTE In t... |
11.7-5 looks like a parabola—and it is, as you can verify by eliminating the parameter t (see Exercise 40). The y-coordinate of the vertex is the maximum height of the ball. It can be found graphically by using trace and zoom-in, or algebraically as follows. The vertex occurs halfway between the two x-intercepts at x ... |
v leads to the following conclusion. 26° by u, Projectile Motion When a projectile • is fired from the position (0, k) on the positive y-axis at an angle with the horizontal, U • in the direction of the positive x-axis, • with initial velocity v feet per second, • with negligible air resistance, then its position at t... |
the center C of the circle at (0, 3). As the circle rolls along the x-axis, the segment rotates through an angle of t radians, as shown in Figure 11.7-10. CP 12 3 3t Figure 11.7-10 y O 3 The distance from T to the origin is the length of the arc of the circle from T to P. From the formula for arc length in Section 6.3... |
5 cos 3t, y 6 sin t 5 sin 3t, 0 t 2p 10. x 3t 2 10, y 4t 3, t any real number 11. x 12 cos 3t cos t 6, y 12 cos 3t sin t 7, 0 t 2p 12. x 2 cos 3t 6, y 2 cos 3t sin t 7, 0 t 2p 13. x t sin t, y t cos t, 0 t 8p 14. x 9 sin t, y 9t cos t, 0 t 20 15. x t 3, y 2t 1, t 0 16. x t 5, y 1t, t 0 17. x 2 t 2, y 1 2t 2, for any t... |
line segment from 6, 12 2 1 30. line segment from (18, 4) to 1 to 12, 10 1 16, 14 2 In Exercises 31–34, locate all local maxima and minima (other than endpoints) of the curve. 31. x 4t 6, y 3t˛ 2 2, 10 t 10 32. x t˛ 3 sin t 4, y cos t, 1.5 t 2 33. 34. x 4t˛ 3 t 4, y 3t˛ 2 5, 2 t 2 x 4t˛ 3 cos t 5, y 3t˛ 2 8, 2 t 2 35.... |
path. Experiment graphically with different angles to find the smallest angle and x 150 b. Use algebra and trigonometry to find the angle needed for the ball to travel exactly 150 feet. y 0. Hint: The ball lands when Use this fact and the parametric equations for the ball’s path to find two equations in the variables ... |
has measure, verify that angle t p 2 and that u in the figure Section 11.7 Plane Curves and Parametric Equations 765 t p x OT CQ 3t 3 cos 2 R Q t p y CT PQ 3 3 sin 2 R Q. b. Use the addition and subtraction identities for sine and cosine to show that in this case x 3. and y 3 1 cos t t sin t 1 2 1 2 y P (x, y 3t 45. a... |
are they closest? d. Confirm your answers in part c as follows. 1 Explain why the distance between particles A and C at time t is given by 8 cos t 3t d 2 2. 2 2 A and C will collide if at some time. Using function graphing mode, graph this distance function when. Zoom-in if necessary, and show that d is always positiv... |
direction from the point (7, 1), as shown in Figure 11.7.A-1. Another parameterization is given by x 3 cos 2t 4 and y 3 sin 2t 1, 0 t p Verify that this last parameterization traces out the circle in a clockwise direction twice as fast as the parameterization given in [1], because t runs from 0 to rather than to 2p. p... |
bolas The hyperbola centered at (c, d) with equation 2 1 x c 2 a˛ 2 1 2 y d 2 b˛ 2 1 can be obtained from the following parameterization. By a Pythagorean identity, x a sec t c y b tan t d, 0 t 2p 1 tan2 t sec2 t. a sec t c c a2 2 y d b2 b tan t d d b2 Therefore a2 1 a sec t a2 2 1 b tan t b2 2 Parametric Equations of ... |
1–4, find a parameterization of the given curve. Confirm your answer by graphing. 1. circle with center (9, 12) and radius 5 2. 2 14x 8y 29 0 2 y˛ x˛ in Section 11.4. Hint: see Example 2 3. 2 y˛ x˛ 2 4x 6y 9 0 4. circle with center 7, 4 1 2 and radius 6 In Exercises 5–26, find parametric equations for the curve whose ... |
........ 701 Characteristics of hyperbolas............... 702 Applications of hyperbolas................. 705 Parabola: focus, directrix, vertex, axis......... 709 Equation of a parabola with vertex at the origin.............................. 710 Characteristics of parabolas................ 711 Applications of parabola... |
..................... 745 Alternate definition of conic sections......... 747 Polar equations of conic sections............ 749 770 Chapter Review 771 Section 11.7 Plane curves............................ 755 Parameter.............................. 755 Parametric equations...................... 755 Eliminating the para... |
the xy term in rotate the axes through an angle u such that Ax2 Bxy Cy2 Dx Ey F 0, cot 2u A C. B The rectangular and polar coordinates of a point are related by x r cos u r2 x˛ 2 y2 and and y r sin u tan u y x 772 Chapter Review If e and d are constants with tion of the form e 7 0, then the graph of a polar equa- r ed... |
of the equation and identify the conic. If there are asymptotes, give their equations and label all characteristic points. Chapter Review 773 19. 21 16 2 2 y 5 4 2 1 1 23. 4x2 9y2 144 20. 3x2 1 2y2 22. 2 1 y 4 25 2 1 2 x 1 4 2 1 24. x2 4y2 10x 9 0 25. 27. 2 6 2y 4 x 3 1 x y˛2 2y 2 2 26. 28. 2 9 x 1 3y 6˛1 y x˛2 2x 3 2... |
. x˛2 3xy y˛2 212x 212y 0 44. x˛2 2xy y˛2 412y 0 45. x˛2 xy y˛2 6 0 774 Chapter Review Section 11.4.A In Exercises 46–47, find the rotation equations when the x- and y-axes are rotated through the given angle. 46. 45° 47. 60° In Exercises 48–49, find the angle through which the x- and y-axes should be rotated to elimin... |
ices (4, 0) and 6, p 1 2 Chapter Review 775 In Exercises 74–77, find a viewing window that shows a complete graph of the curve with the given parametric equations. 74. x 64 cos 3 p6 1 2 4 t and y 16t2 64 sin S p 6 T t, 0 t p 75. x t˛ 3 t 1 and y t2 2t, 3 t 3 76. x t2 t 3 and y t3 5t, 3 t 3 77. x 8 cos t cos 8t and y 8 ... |
1 89. 4x2 9y2 1 91. x2 36y2 1 93. 95. 2 1 x 2 49 2 1 2 y 5 64 12 2 1 97. x 32 11 Figure 11.C-1 Arc Length of a Polar Graph Many applications of calculus involve finding the distance along a curve, or arc length. Although calculus is usually needed to find the exact value of the arc length, approximations are often suf... |
2p 3 R 2 4p 3 R 2 2 4p 3 R Q 2 2p 2 1 2p 3 R Q 4p 1 3 R Q 4p 3 R Q 3.63 5.54 2p 2 Segment from u p to Segment from Segment from u 4p 3 u 5p 3 : u 4p 3 u 5p 3 2 2p 2 B 1 2 8p 3 b a 2p 1 2 a 8p 3 b 7.55 : B a 2 8p 3 b 10p 2 a 3 b 8p 3 b a a 10p 3 b 9.60 to to u 2p : 10p 2 B a 3 b 2 4p 2 1 10p 3 b 1 a 4p 2 11.66 The appr... |
12.2 > 12.3 > 12.4 > 12.5 Real-world situations often require a common solution to several equations with multiple variables. Such a collection of equations is known as a system of equations. Solutions to a system of equations in two or three variables may be represented geometrically by intersections of lines or plan... |
solution of the system. The set of values is a solution of the first two equations, but not the third, so it is not a solution of the system. z 12 x 0, Solutions of systems of equations in two variables can be found numerically by comparing tables of values for the equations. Example 1 Solving a System Numerically Fin... |
there are exactly three geometric possibilities. • the lines can be parallel and have no point of intersection • the lines can intersect at a single point • the lines can coincide Each of these possibilities leads to a different number of solutions for the system. The three types of linear systems are shown below. 2 2... |
by graphing, as shown in Figure 12.1-4, where y 1.64. x 3.45 and Elimination Method Elimination is another algebraic method used to solve systems. To solve a system using the elimination method: 1. Multiply one or both of the equations by a nonzero constant so that the coefficients of x (or y) are opposites of each ot... |
1-6 Section 12.1 Solving Systems of Equations 785 6x 12y 18 6x 12y 18 0 0 0 0, The last equation, is always true. This indicates that the two equations represent the same line, and every ordered pair that satisfies the first equation must also satisfy the second equation. Thus, the system has infinitely many solutions.... |
1] from the original system, and solve. x 2 and y 1 into equa The solution is x 2, y 1, z 4. The solution should be checked in all equations of the original system. ■ Applications of Systems Systems of equations occur in many real-world applications. The simplest situations involve two quantities and two linear relatio... |
price of blend price per pound q r q weight of coffee r price per pound r weight of coffee q q r 4.50x 7.00y $5.00 788 1 Chapter 12 Systems and Matrices Solve the system of equations. x y 1 4.5x 7y 5 0 1 Figure 12.1-8 Multiply the first equation by and add it to the second equation. 7 7x 7y 7 4.5x 7y 5 2.5x 2 x 0.8 1 ... |
42x 56y 28 25. 9x 3y 1 6x 2y 5 27. x 3 2x 5 y 2 y 5 3 2 18. 4x 3y 1 x 2y 19 20. 2x 5y 8 6x 15y 18 22. 2x 8y 2 3x 12y 3 24. 1 y 1 x 2 6 5 3 20x 24y 10 26. 8x 4y 3 10x 5y 1 28. x 3 x 6 3y 5 y 2 4 3 29. 30 2y 3 x y 3 3x y 2 1 2 31. 3.5x 2.18y 2.00782 1.92x 6.77y 3.86928 32. 463x 80y 13781.6 0.0375x 0.912y 50.79624 33. 34... |
equations in part a. c. Where do the lines intersect? What is the significance of this point? d. Which parcel will be worth more in five years? in 15 years? 39. A toy company makes dolls, as well as collector cases for each doll. To make x cases costs the company $5000 in fixed overhead, plus $7.50 per case. An outsid... |
remain accurate, when will the death rates for heart disease and cancer be the same? (Source: U.S. Department of Health and Human Services) 45. At a certain store, cashews cost $4.40 per pound and peanuts cost $1.20 per pound. If you want to buy exactly 3 pounds of nuts for $6.00, how many pounds of each kind of nuts ... |
usually called the x-axis, the y-axis, and the z-axis. In three-dimensional coordinates, the arrowhead on each axis indicates the positive direction. Each pair of axes determines a coordinate plane, which is named by the axes that determine it. There are three coordinate planes, the xy-plane, the yz-plane, and the xz-... |
direction slope in y-direction z-intercept general form of a plane point-slope form of a plane yz-plane xz-plane xy-plane plane: parallel to yz-plane parallel to xz-plane parallel to xy-plane Graphing Planes One method of graphing a plane in three dimensions is to find the x-, y-, and z-intercepts, plot the intercepts ... |
) x-intercept (−1, 0, 0) z-intercept Plot the intercepts and sketch the plane that contains them, as shown in Figure 12.1.A-6. Figure 12.1.A-6 y ■ y ■ Graphical Representations of 3 3 Systems A linear system of equations in three variables is represented graphically 3 3 by three planes. A solution of a linear system i... |
rices • Solve application s using systems. It is often convenient to use an array of numbers, called a matrix, as a method to represent a system of equations. For example, the system is written in matrix form as x 2y 2 2x 6y 2 1 2 a 2 6 2 2b In this shorthand, only the coefficients of the variables are written. This re... |
below by using elementary row operations in the elimination method on the left and the augmented matrix method on the right. Compare the steps performed in each method. x 2y 2 2 6y 2 2b 2x 1 2 2 6 a NOTE In previous methods, the step to replace a row with the sum of itself and a multiple of another row was done in two... |
matrix, find the solutions, if any, and classify each system as consistent, consistent with infinitely many solutions, or inconsistent. a. 1 0 a 3 0 4 0b Solution b. 1 0 a 2 0 1 3b a. The system represented by the augmented matrix is x 3y 4 0x 0y 0 798 Chapter 12 Systems and Matrices The second equation,, is always tr... |
4 Write the augmented matrix for the system, enter the matrix into a calculator, then reduce to reduced row-echelon form, as shown in Figure 12.2-2. The last row of the reduced matrix represents the equation 0x 0y 0z 1. Figure 12.2-2 Because the equation has no solution, the original system has no solution and is ther... |
on the 9% card. The total amount borrowed is $10,000. x y z 10,000 Total interest is the sum of the amounts of interest on the three cards. Interest on 18% card 0.18x Interest on 15% card 0.15y Interest on 9% card 0.09z Total interest 1244.25 The amounts on the cards are related by a third equation. Amount on 15% card... |
13. 15. 17. 19. x 3y 2z 0 2x 3y 2z 3 x 2y 3z 0 x 2y 2z 1 x 2y 2z 4 2x 2y 3z 5 x 2y 4z 6 x y 13z 6 2x 6y z 10 x y z 200 x 2y 2z 0 2x 3y 5z 600 2x y z 200 14. 16. 18. 3x 7y 9z 0 x 2y 3z 2 x 4y z 2 2x y 2z 1 3x y 2z 0 7x y 3z 2 x y 5z 6 3x 3y z 10 x 3y 2z 5 20. 3x y z 6 x 2y z 0 In Exercises 21–35, solve the system by an... |
coordinates in the second column, and the z-coordinates in the third column. Each row represents a point. A crystal lattice is used to represent the atomic structure of a crystal. The two matrices below represent simple cubic and A10 crystal lattices, in which the atoms of the crystal are at the points represented by t... |
investor has $70,000 invested in a mutual fund, bonds, and a fast food franchise. She has twice as much invested in bonds as in the mutual fund. Last year the mutual fund paid a 2% dividend, the bonds 10%, and the fast food franchise 6%; her dividend income was $4800. How much is invested in each of the three investme... |
hours available each week in the cutting department, 1490 hours in the assembly department, and 2160 in the finishing department. Manufacturing a chair requires 0.2 hours of cutting, 0.3 hours of assembly, and 0.1 hours of finishing. A chest requires 0.5 hours of cutting, 0.4 hours of assembly, and 0.6 hours of finish... |
columns in the matrix. An matrix has m rows and n columns. For example matrix 3 rows, 3 columns B 3 2 5 0 12 4 1 § matrix ¥ 4 rows, 1 column Each entry of a matrix can be located by stating the row and column in is the entry in row i and column j of its which it appears. An entry 5 corresponding matrix. In the matrice... |
find 3A. 806 Chapter 12 Systems and Matrices Solution Multiply each entry of A by 3. 3A 18 6 6 3 0 15 21 15 ¢ Figure 12.3-2 The results are confirmed in Figure 12.3-2. ■ Matrix Multiplication To multiply two matrices, multiply the rows of the first matrix by columns of the second matrix. The number of entries in each ... |
2 4b ° 2 0 1 3 5 8 ¢ row 1 of A column 1 of B ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ row 1 of A column 2 of B ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ row 2 of A column 1 of 21 row 2 of A column 2 of B 8 8 1 1 2 2 b ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ 8 2 a 12 35b BA is a 3 3 matrix. BA 4b ° 21 21 21 20 34 ¢ Note that AB BA. CAUTION in general. AB BA Matrix multiplication is not commu... |
2.5 hr 4 hr 7 hr finishing 1 hr 1.5 hr 4.5 hr ¢ removing finish sanding finishing cost per hour $7 $18 $10 ° ¢ Figure 12.3-5 Section 12.3 Matrix Operations 809 Solution 3 3 so multiplication of the The first matrix is and the second is 3 1 first matrix by the second matrix is defined, and the product is a matrix. In t... |
S M S M ■ Example 6 Food Webs A food web shows the relationships between certain predators and prey in an ecosystem. A directed network can be used to represent a food web, with the arrows pointing in the direction of prey to predator. Write an adjacency matrix for the following food web. NOTE Two different two-stage ... |
sheet cake 3 -tiered cake 0.5 hr 0.75 hr 1.5 hr ° 0.25 hr 0.5 hr 1.25 hr ¢ In Exercises 19–22, show that AB is not equal to BA by computing both products. The cost per hour for baking and decorating is given by the matrix below. Find the product of the two matrices and interpret the result. 19. A 3 5 a 2 1b 20 6b B 3 ... |
the result. 34. A delivery company ships packages between certain locations, as shown by the directed network below. mix A mix B nuts fruit 30% a70% 45% 55%b Dallas/Fort Worth Amarillo 31. Write an adjacency matrix for the food web represented by the directed network below. El Paso Austin Houston cheetahs gazelles lio... |
. One method of solving similar equations with real numbers is by multiplying both sides of the equation by the inverse of a. AX B, ax b 1ax a a x a 1b 1b [2] The solution of equation [2] depends on the fact that which is the identity for multiplication of real numbers. Thus, in order to define the inverse of a matrix,... |
u. vb Then AA 1 I2. 1 AA 2 1 a 6 4b a x y u vb a 2u 6v u 4vb 1 0 a 0 1b I2 2x 6y x 4y 1 Setting the corresponding entries of in two systems of equations, one for each column. AA I2 and equal to each other results 2x 6y 1 x 4y 0 2u 6v 0 u 4v 1 The solutions of the two systems are x 2, y 1 2 and u 3, v 1. Thus, 1 A 2 1 ... |
System Figure 12.4-2 818 Chapter 12 Systems and Matrices Example 6 Solving a 3 3 System Using a Matrix Equation Use an inverse matrix to solve Solution The coefficient matrix is A B °. Then X A 1B ¢ 2 5 1 x y z 2 2x 3y 5 x 2y z 1 1 0 1 ¢ and the constant matrix is, so the solution is x 7, y 3, as shown in Figure 12.4-... |
a 2 5 3 4 1 4 1 3b 1 7 ¢ In Exercises 9–12, write a set of systems of equations that represent the solution of the matrix equation AA1 In. (See Example 4.) Do not solve the systems. 9. A 2 4 a 0 1b 10. A 1 2 a 3 5b 11 ¢ 12 ¢ In Exercises 13–20, find the inverse of the matrix, if it exists. 13. 1 3 a 2 4b 15. 3 6 a 1 2... |
5v 3w 0 x 2y z 2v 4w 0 35. Critical Thinking Consider the two systems of equations below. x 2y 3 3x 6y 9 x 2y 4 3x 6y 7 a. Write a matrix equation that represents each system. Which matrices in the two equations are the same? Does either matrix equation have a solution? b. Solve each system by any method. Make a conje... |
b. Use this equation to estimate the CO2 concentration in the years 1983, 1993, and 2003. For comparison purposes, the actual concentrations in 1983 and 1993 were 343 ppm and 357 ppm respectively. 45. Find constants a, b, and c such that the points (0, 2) (ln 2, 1), and (ln 4, 4) lie on the graph of f. (See Example 7.... |
of mix S, and 44,000 units of mix T are available each day, how many of each type of animal can be supported? Section 12.5 Nonlinear Systems 821 12.5 Nonlinear Systems Objectives • Solve nonlinear systems algebraically • Solve nonlinear systems graphically The matrix methods discussed in Section 12.2 and Section 12.4 ... |
equations. Notice that if the values for x were substituted into the first equation y ;7 This would instead, the resulting solutions would be or 29, 41 29, 41, and give 4 solutions, (5, 7), However, the 5, 7 do not satisfy the second equation; they solutions are extraneous. 5, 7, 2 29, 41 y ;41. and Thus, the solution... |
. 3, 2 There are four solutions: (3, 2), ( ). The first two are exact solutions, a fact that can be confirmed by substituting the values into the original equations. 2.2, 2.6 2.2, 2.6 ), and ( ), ( ■ 824 Chapter 12 Systems and Matrices Example 5 Application of a Nonlinear System The revenue and cost (in dollars) for ma... |
y 0.25x4 2x2 4 y x3 x2 2x 1 17. y x3 x 1 y sin x 18. y x2 4 y cos x 19. 25x2 16y2 400 9x2 4y2 36 20. 9x2 16y2 140 x2 4y2 4 21. 5x2 3y2 20x 6y 8 x y 2 22. 4x2 9y2 36 2x y 1 23. x2 4xy 4y2 30x 90y 450 0 x2 x y 1 0 24. 3x2 4xy 3y2 12x 2y 7 0 x2 10x y 21 0 25. 4x2 6xy 2y2 3x 10y 6 4x2 y2 64 26. 5x2 xy 6y2 79x 73y 196 0 x2... |
is to be rolled into a circular tube. If the tube is to have a surface area (excluding ends) of 210 square inches and a volume of 252 cubic inches, what size of metal sheet should be used? (Recall that the circumference of a circle with radius r is that the volume of a cylinder with radius r and height h is pr2h. 2pr ... |
is the solution, choose a point that is not on the line, such as (0, 0), and test it in the inequality.? 2 0 2 0 1 2 False The inequality is false for the test point, so shade the region that does not contain that point—in this case, the region above the line. ■ Figure 12.5.A-1 The method used in Example 1 can be summ... |
77 mx b is the half-plane above the line. • The solution of y 66 mx b is the half-plane below the line. y mx b For and part of the solution. y mx b, the line y mx b is also Solving Linear Inequalities in Two Variables 828 Chapter 12 Systems and Matrices Technology Tip To display shading on TI models, select the graph ... |
5. The first figure is bounded on all four sides and the two other figures have one side that is not bounded. P Q S R Figure 12.5.A-5 A corner point of such a region is any point where two of the sides intersect, such as points P, Q, R, and S in the first region in Figure 12.5.A-5. The key to solving linear programming... |
.5 24 2 2 2 2 The solution is the corner point (6, 3), which yields the largest value of the objective function, 24. This is the maximum value of the objective function. ■ Example 5 Application Carla is making earrings and necklaces to sell at a craft fair. The profit from each pair of earrings is $3, and the profit fr... |
or least) value of the objective function. This is the maximum (or minimum) value of the function on the feasible region. Exercises 12.5.A In Exercises 1–12, solve the system of inequalities. 1. 3. y 2x 4 y 7 x 2 y 7 3x 1 y 6 1 x 3 2 2. y x 3 y 4x 2 4. y 1 x 1 4 y 7 4x 1 5. 7. 2x 3y 7 6 x 2y 5 7 2x 1 y x 6. 8. 2x 4y 5 ... |
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