Answers to All Exercises. Appendix C ( ) ( ) Section C.1 (page C7) APPENDICES. Answers to All Exercises Ans1

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1 Answers to All Eercises Ans Answers to All Eercises Appendi C Section C. (page C). Cartesian. Distance Formula. Midpoint Formula. ( h) + ( k) = r, center, radius. c. f. a. d. e. b. A: (, ); B: (, ); C: (, ); D: (, ). A: (., ); B: (, ); C: (,.); D: (, )... (, ) (, ) (, ) (, ) (,.) (., ) (, ) (, ). (, ) (, ) (, ) ( ), (, ) ( ), (, ) (, ). (, ). (, ). (, ). (, ). Quadrant IV. Quadrant III. Quadrant II. Quadrant I. Quadrant III or IV. Quadrant I or IV. Quadrant III. Quadrant III. Quadrant I or III. Quadrant II or IV (a),, (b) + =. (a),, (b) + =. (a),, (b) + = ( ). (a),, (b) + = ( ). ( ) + ( ) = ( ). ( ) + ( ) = ( ). Two equal sides of length. Two equal sides of length. Opposite sides have equal lengths of and.. Opposite sides have equal lengths of and.. The diagonals are of equal length ( ). The slope of the line between (, ) and (, ) is. The slope of the line between (, ) and (, ) is. The slopes are negative reciprocals, indicating perpendicular lines, which form a right angle.. The diagonals are of equal length ( ). The slope of the line between (, ) and (, ) is. The slope of the line between (, ) and (, ) is. The slopes are negative reciprocals, indicating perpendicular lines, which form a right angle.. (a) (b) (c) (, ). (a). (a). (a). (a) (, ) (, ) (, ) (b) (, ) (, ) (, ) (, ) (, ) (, ) (c) (, ) (b) (c) (, ) (b) (, ) (c) (, ) (b) (c) (, ) APPENDICES

2 Ans Answers to All Eercises. (a) (, ) (b) (c) (, ). Center: (, ). Center: (, ) Radius: Radius:. (a) (, ) (, ) (, ) (b) (c) (, ). Center: (, ). Center: (, ) Radius: Radius:. (a) (b) (c) (, ). Center: (, ). Center: (, ) (, ) Radius: Radius:. (a). (a) (, ) (.,.) (.,.) (.,.) (.,.) (b). (c) (.,.) (b). (c) (.,.). $. million. $. million. + =. + =. ( ) + ( + ) =. ( + ) + ( ) =. ( + ) + ( ) =. ( ) + ( + ) =. ( ) + ( ) =. + =. ( + ) + ( ) =. ( ) + ( + ) =. ( ) + ( + ) =. ( + ) + =. ( ) + ( + ) =. ( + ) + ( ) =. (, ), (, ), (, ). (, ), (, ), (, ), (, ). (, ), (, ), (, ), (, ). (, ), (, ), (, ). d.. ft;. ft.. km. True. The lengths of the sides from (, ) to (, ) and from (, ) to (, ) are both.. False. It could be a rhombus.. False. You would have to use the Midpoint Formula times.. ;. No. The scales depend on the magnitudes of the quantities measured.. ( m, m ); (a) (, ) (b) (, ). ( +, + ( +, + ) ), ( +, + ), (a) (, ), (, ), (, ) (b) (, ), (, ), (, ). Proof. (a) ii (b) iii (c) iv (d) i

3 Answers to All Eercises Ans Appendi C. (page C). solution point. graph. Algebraic, graphical, numerical.. If possible, rewrite the equation so that one of the variables is isolated on one side of the equation.. Make a table of values showing several solution points.. Plot these points on a rectangular coordinate sstem.. Connect the points with a smooth curve or line.. (a) Yes (b) Yes. (a) Yes (b) Yes. (a) No (b) Yes. (a) No (b) Yes. (a) No (b) Yes. (a) No (b) Yes. (a) Yes (b) No. (a) Yes (b) No. Solution point (, ) (, ) (, ) (, ) (, ). Solution point (, ) (, ) (, ) (, ) (, ). b. d. c. a. e. f... Solution point (, ) (, ) (, ) (, ) (, ).. APPENDICES... Solution point (, ) (, ) (, ) (, ) (, )..

4 Ans Answers to All Eercises.... Intercepts: (, ), (, ) Intercepts: (, ), (, ) Intercepts: (, ), (, ), (, ). Xmin = - Xma = Xscl = Ymin = - Yma = Yscl =. Xmin = - Xma = Xscl = Ymin = - Yma = Yscl =. The graphs are identical. Distributive Propert. The graphs are identical. Associative Propert of Addition. The graphs are identical. Associative Propert of Multiplication. The graphs are identical. Multiplicative Inverse Propert. (a) (,.) (b) (, ). Intercepts: (, ), (, ), (, ).. (a) (, ) (b) (., ). Intercept: (, ) Intercept: (, ).. (a) (.,.) (b) (., ), (., ), (., ). Intercepts: (, ), (, ) Intercepts: (, ), (, ).. (a) (, ) (b) (.,.), (.,.), (.,.), (.,.). Intercepts: (, ), (, ) Intercepts: (, ), (, ).. =. = = =. Intercepts: (, ), (, ), Intercepts: (, ), (, ), (, ) (, ).. = + ( ). = + ( ) = ( ) = ( ) Intercepts: (, ), (, ) Intercepts: (, ), (, ). b, c, d. c, d

5 Answers to All Eercises Ans. (a),. (a) (b) $, (b). r (c). r (c) $.. (a) Year New houses (in thousands).... Year New houses (in thousands) The model fits the data well. (b).... The model fits the data well. (c) :,; :,; Yes. Answers will var. (d),. (a) The model fits the data well. (b).; The life epectanc in (c) t = or (d) About. r. False. = has two -intercepts.. False. = has an infinite number of -intercepts.. Use the equations = +. and = +. to model the situation. When the sales level equals $,, both equations ield $. Appendi C. (page C). equation. solve. etraneous. point of intersection. (a) Yes (b) No (c) No (d) No. (a) No (b) Yes (c) No (d) No. (a) Yes (b) No (c) No (d) No. (a) No (b) No (c) Yes (d) No. (a) No (b) No (c) No (d) Yes. (a) No (b) Yes (c) No (d) No. Identit. Identit. Identit. Identit. Conditional equation. Conditional equation. =. =. =. =. =. z =. =. =. z =. =. z =. =. u =. =. =. =. =. =. =. No solution. =. z =. No solution. No solution. (, ), (, ). (, ), (, ). (, ), (, ), (, ). (, ), (, ), (, ). (, ), (, ). (, ), (, ). No intercepts. (, ). (, ), (, ), (, ). (, ), (, ), (,.). (, ), (, ). (, )... (, ) (, ). (, ) (, ). ( ) =. ( ) + = APPENDICES = +. = +.,,. () () + () =. () () + () = () () + () = () () + () = () () + () = () () + () =. (a) Xmin = - Xma = Xscl = Ymin = - Yma = Yscl = (b) (, ), (, ), (, ) (c) (, ): No (, ): Yes Answers will var.. + =

6 Ans Answers to All Eercises. () = =. (a) (b) and (c) =,, (d) The are the same ,...,..,....,,. ±,. ±..,... ±.,.,... (, ). (, ). (, ), (, ). (, ), (, ). (, ). (, ). (, ). (, ). (.,.), (.,.). (.,.). (, ), (, ), (, ). (, ), (, ).,. ±.,.,.,.,.,.,. a ± b. a. ±. ±.,.,. No solution. No solution. ± ;.,. ±. ;.,... ;..,.,. ±. ±. ±.,. No solution. ± ±.. ±. ±. ±. ±. ±. No solution. No solution... ±. ±. ±.,.,. No solution.. No solution. ±. ±.. ± b a. ±,. ±,.,.,., ±., ±.,,. ±, ± i. ±, ±. ±. ±, ±. ±.,..,.,.... No solution ,.,, ±..,, ±.. ±.,.,.,.,.,.,. (a). (a). (a). (a). (a). (a). (a). (a) (b) and (c) = ±, ± (d) The are the same. (b) and (c) =, (d) The are the same. (b) and (c) = (d) The are the same. (b) and (c) = (d) The are the same. (b) and (c) = (d) The are the same. (b) and (c) =, (d) The are the same. (b) and (c) =, (d) The are the same. (.,.); In, both states had the same population. (b) (.,.); In, both states had the same population. (c) Change in population per ear; Marland s population is growing faster. (d) Marland:,,; Wisconsin:,, Answers will var.

7 Ans Answers to All Eercises. (a) (b) Answers will var. (c). < <. <.. (d) (e) Answers will var.. (a) >.. (b). C (c).. (a) < >.. (b). F (c).. False. The lines could be identical.. False. See Eample.. c =. c = b. (a), (b), a lb in. Appendi C. (page C). <.... >. <. <. < <.,. <, >. < <...,. <. <. (a) (a) (b) (b). <, >.. > APPENDICES (a) (b) (a) (b)... double. a a. a, a. zeros, undefined values. No. Transitive Propert. d. a. f. b. e. c. (a) Yes (b) No (c) Yes (d) No. (a) No (b) No (c) Yes (d) Yes. (a) No (b) Yes (c) Yes (d) No. (a) Yes (b) Yes (c) Yes (d) No. >..... (a) (a) (b), (b),. >. + <. <. + + >. Positive on: (, ) (, ) Negative on: (, )

8 Ans Answers to All Eercises. Positive on: (, ) (, ) Negative on: (, ). Positive on: (, ) ( +, ) Negative on: (, + ) ). Positive on: (, + Negative on: (, ) ( +, ). Negative on: (, ). Positive on: (, ). (, ], [, ). (, ). (, ). (, ], [, )., [, ). (, ]. (, ), (, ). (, ), (, ). (, ), (, ). [, ], [, ). No solution.. (, ), (, ). All real numbers. (a) = (b) (c) >. (a) =, (b), (c) <, >.. (a), (a) <, (b) (b) =, <. (, ), (, ). (, ), (, ). (, ), [, ). (, ].. (a) < (a) (b) < (b) <. [, ). [, ). [, ]. (, ], [, ). (, ], [, ). (, ). (a) (b) (, ); (, ). (a) (b) (, ); (, ). (a) sec (b) (, ). (a) sec (b) (,.), (., ). (a) (b) (, ) (c). < <.. (a) and (b) (c). lb (d) Answers will var.. t.; Starting in, there were at least Bed Bath & Beond stores.. t., t.; From to and to, there were at most Williams-Sonoma stores.. t.; In, there were about the same number of Bed Bath & Beond stores as Williams-Sonoma stores.. t.; Starting in, the number of Bed Bath & Beond stores eceeded the number of Williams-Sonoma stores.. vibrations sec.. mm.. < t <.. < v <. False.. True. + + is alwas positive.. False. Cube roots have no restrictions on the domain.. iv, ii, iii, i. (a) iv (b) ii (c) iii (d) i. (a) a, b (b) Positive on: (, a) (b, ) Negative on: (a, b) When < a or > b, the factors have the same sign so the product is positive. When a < < b, the factors have opposite signs so the product is negative. (c) = a and = b Appendi D (page D). directl proportional. constant, variation. directl proportional. inverse. combined. jointl proportional. =. =. =. =. Model: = ;. cm,. cm. Model: = ;. L,. L

9 Answers to All Eercises Ans. =.; $. =.; $.. (a). m (b) N. N.... = k = k = k = k.. mi/h. k(v) kv =.... = k = k = k = k. =. =. =. =. A = kr. V = ke. = k k s. h = s. F = kg r. z = k. P = k V. R = ks(l S). R = k(t e T). F = km m r APPENDICES

10 Ans Answers to All Eercises. The area of a triangle is jointl proportional to its base and height.. The surface area of a sphere varies directl as the square of its radius.. The volume of a sphere varies directl as the cube of its radius.. The volume of a right circular clinder is jointl proportional to the product of its height and the square of its radius.. Average speed is directl proportional to the distance and inversel proportional to the time.. ω varies directl as the square root of g and inversel as the square root of W.. A = πr. =. =. z =. F = rs. P =. z =. v = pq s.. ft.. in.. J. No. The -inch pizza is the best bu.. The velocit is increased b.. (a) The safe load is unchanged. (b) The safe load is eight times as great. (c) The safe load is four times as great. (d) The safe load is one-fourth as great.. (a) C Temperature (in C) Depth (in meters) (b) Yes. k =, k =, k =, k =, k = (c) C = d (d) (e) About m d. Inversel. Directl Appendi E (page E). linear. equivalent inequalities No solution. All real numbers..... <. >. <. <... >. <. <. >. <. >..... <... <.. > Appendi F Appendi F. (page F). solution. graph. linear. point, equilibrium. g. d. a. h. e. b. f. c (a) Length (in centimeters) Force (in pounds) (b) Yes. (c). lb. False. will increase if k is positive and will decrease if k is negative.. False. E is jointl proportional to the mass of an object and the square of its velocit. F....

11 Ans Answers to All Eercises (, ) (, ).. (, ) (, ). (, ) (, ) (, )... + >.. +. >. (a) Yes (b) No (c) No (d) No. (a) No (b) No (c) No (d) Yes.. APPENDICES.. (, ) (, ) (, )

12 Ans Answers to All Eercises (, ) (, ) No solution ( ),.. (, ) (, ) No solution. (, ). (, ) (, ) (, ( ) + ) (, ) (, ). (, ) (, ) (, ) (, ). (, ) (, ) (, ) (, ).. (, ) (, ) ( ), { + <.. {. { + <. + ( ) + <. + >. {... { + Consumer surplus. Producer surplus (, ) Consumer surplus: Producer surplus:.. Consumer surplus (, ) Producer surplus Consumer surplus: Producer surplus:, Consumer surplus Producer surplus, (,,, { + + Consumer surplus:,,. Producer surplus:,,. ( (, e ) (, e ) (, ) (, ). { {

13 Answers to All Eercises Ans. Consumer surplus Producer surplus (,,, ),,,,. (a) { + (b), Consumer surplus:,, Producer surplus:,,. (a) { +, (b),,,. (a) { + π (b),. (a) + { + (b),,,. (a) {π π > > (b). (a) (b),,, { + + +, (c) The line is an asmptote to the boundar. The larger the circles, the closer the radii can be and still satisf the constraint.. True. False. + is outside the parabola.. Test a point on either side.. Answers will var. Appendi F. (page F). optimization. objective function. constraints, feasible solutions. The vertices. Minimum at (, ):. Minimum at (, ): Maimum at (, ): Maimum at (, ):. Minimum at (, ):. Minimum at (, ): Maimum at (, ): Maimum at (, ):. Minimum at (, ):. Minimum at (, ): Maimum at (, ): Maimum at (, ):. Minimum at (, ):. Minimum at (, ): Maimum at (, ): Maimum at (, ):. Minimum at (, ): Maimum at (, ):. Minimum at (, ):, Maimum at (, ):,. Minimum at (, ): Maimum at an point on the line segment connecting (, ) and (, ):. Minimum at (, ):, Maimum at (, ):, APPENDICES

14 Ans Answers to All Eercises.. (, ) (, ). (, ) Minimum at an point on the line segment connecting (, ) and (, ): Maimum at (, ):. (, ) (, ) (, ) (, ) Minimum at (, ): Minimum at (, ): Maimum at (, ): Maimum at (, ):.. (, ) (, ) (, ) Minimum at (, ): Maimum at (, ):. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Minimum at (, ): Minimum at (, ): Maimum at (, ): Maimum at (, ):. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Minimum at an point on the line segment connecting (, ) and (, ): Maimum at (, ): (, ) (, ) (, ) (, ). Minimum at (, ): Maimum at (, ):.. Minimum at (, ): Maimum at (, ): (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Minimum at (, ): Maimum at an point on the line segment connecting (, ) and (, ): Minimum at an point on the line segment connecting (, ) and (, ): Maimum at (, ):.. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Minimum at an point on the line segment connecting (, ) and (, ): Maimum at (, ): Minimum at an point on the line segment connecting (, ) and (, ): Maimum at (, ):

15 Answers to All Eercises Ans. Minimum at (, ): Maimum at (, ):. (, ) (, ) (, ) (, ) (, ) (, ). (a) and (b) (c) (, ) + =. (, ) (, ) The constraints do not form a closed set of points. Therefore, z = + is unbounded. (, ) = + (, ) + = (, ). (a) and (b) (c) (, ) (, ) + = (, ) = + (, ) + =. (a) and (b) (c) (, ) (, ) (, ) + = + = = + (, ). (a) and (b) (, ) + = (, ) (, ) + = (c) Maimum at an point on the line segment connecting (, ) and (, ). (, ) ( ) (, ), (, ) z is maimum at an point on the line segment connecting (, ) and (, )... The feasible set is empt. (, ) (, ) (, ) The constraint is etraneous. Maimum at (, ):. (a) Four audits, ta returns (b) Maimum revenue: $,. units of model A; units of model B Maimum profit: $,. (a) Three bags of brand X, si bags of brand Y (b) Minimum cost: $. Three bags of brand X; two bags of brand Y Minimum cost: $. True. True. z = +. z = +. z = +. z = +. (a) t > (b) < t <. (a) < t < (b) t > Appendi G (page G). mathematical induction. first. arithmetic. second.. (k + )(k + ) (k + )(k + ) APPENDICES

16 Ans Answers to All Eercises. k+. k (k + )! (k + )! (k ) + (k + ) (k + ) + (k + ). Answers will var..,,.,..,. Answers will var..,,,, First differences:,,, Second differences:,, Linear.,,,, First differences:,,, Second differences:,, Neither.,,,, First differences:,,, Second differences:,, Quadratic.,,,, First differences:,,, Second differences:,, Neither.,,,, First differences:,,, Second differences:,, Quadratic.,,,,, First differences:,,,, Second differences:,,, Neither.,,,, First differences:,,, Second differences:,, Linear.,,,, First differences:,,, Second differences:,, Quadratic. a n = n n +, n. a n = n n +, n. a n = n + n. a n = n n +. (a) a n = () n (b) a n = [ + n ( k= ) k ] (c) ( ) n. (a) (b) (c) a n = n (d) Answers will var.. False. Not necessaril. False. The first differences are all the same.. False. It has n second differences.. (a) P n is true for all integers n. (b) P n is true for all integers n. (c) P, P, and P are true. (d) P n is true for an positive integer n.