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Name ______________________ MATH 1090 FINAL EXAM instructor: . date Dec. 2008 2 hours with calculator section / time: . Work problems completely, either on this paper, or on another sheet, which you include with this paper. Credit will be given for work, so show all necessary work. If you turn in work on another paper, number the problems so they can be found and read. If you answer “none of the preceding,” tell what the answer should be. Work 15 of the18 problems on this exam. Mark out the 3 problems that will not be graded. The first 15 not marked out will be scored. 1. The resident population of Florida (in thousands) is given in the table at the right. Let x equal the number of years after 1980 and y equal the number of thousands of residents. Identify the linear model that is the best fit for this data. Population Years (thousands) a) y = 168.92x + 10,642.32 b) y = 285.27x + 9875.17 1980 9,746 1985 11,351 c) y = 285.27x – 12,946.37 d) y = 3.5x2 + 350x + 9732 1990 12,938 1995 14,180 e) y = 9970.32(1 + 0.023) x f ) None of the preceding. 1996 14,425 1997 14,677 1998 14,916 2. At the end of an advertising campaign, weekly sales declined according to the equation y = 10,000(3–0.05x) dollars, where x is the number of weeks after the campaign ended. Determine the sales at the end of the campaign, and 8 weeks after the end? a) end $0, 8-weeks $7572 b) end $10,000, 8-weeks $6444 c) end $0, 8-weeks $6444 d) end $10,000, 8-weeks $8452 e) end $10,000, 8-weeks $7572 f ) None of the preceding. 3. The average price of a U.S. movie ticket is given in the Ticket Ticket table. Use a model of this data to make the best estimate Year Year Price ($) Price ($) of when the price was $5.70. 1991 4.21 1996 4.42 1992 4.15 1997 4.59 a) 2000 b) 2001 1993 4.14 1998 4.69 1994 4.18 1999 5.08 c) 2003 d) 2005 1995 4.35 e) 2006 f ) None of the preceding. 2 4. A delivery service has three types of aircraft, each of Aircraft Type which carries three types of Units Carried Passenger (P) Transport (T) Jumbo (J) cargo. The payload of each Next-Day Letters (a) 200 200 200 type is summarized in the Two-Day Letters (b) 300 40 700 table at the right. Suppose Air Freight (c) 40 130 70 on a given day the airline must move 2200 next-day delivery letters, 3860 two-day delivery letters, and 920 units of air freight. Identify the system that tells how many aircraft of each type should be scheduled. 200a 300b 40c 2200 200x 200 y 200z 2200 a) 200a 40b 130c 3860 b) 300x 40 y 700z 3860 200a 700b 70c 920 40 x 130 y 70 z 920 200P 300T 40 J 2200 200a 200b 200c 2200 c) 200P 40T 130J 3860 d) 300a 40b 700c 3860 200P 700T 40a 130b 70c 920 70 J 920 200P 200T 200 J 2200 e) 300P 40T 700 J 3860 f ) None of the preceding. 40 P 130T 70 J 920 5. Suppose a long-distance call costs 99¢ for the first minute plus 25¢ for each additional minute. Identify the equation that gives the cost C of a call lasting n minutes. a) C(n) = 0.99 + 0.25(n – 1) b) C(n) = 0.99 + 0.25n c) C(n) = 0.99 + 0.25n d) C(n) = 0.99 + 0.25(n – 1) e) C(n) = 0.99 + 0.25(n + 1) f ) None of the preceding. 6. A loan of $36,000 is to be amortized with monthly payments over 6 years. If the interest on the loan is 6% per year, paid on the unpaid balance, what payment is required each month to amortize the loan? a) $5925.44 b) $416.62 c) $596.62 d) $6105.44 e) $311.07 f ) None of the preceding. 3 7. Suppose the demand for artificial Christmas trees is given by the function p = 109.70 – 0.10q and that the supply of these trees is given by p = 0.01q2 + 5.91 where p is the price of a tree in dollars and q is the quantity of trees that are demanded/supplied in hundreds. Find the price that gives market equilibrium. a) $91.60 b) $97.00 c) $100.00 d) $107.00 e) $120.40 f ) None of the preceding. 8. The average cost per unit for the production of a certain brand of DVD players is given by 400 50 x 0.01 x 2 C where x is the number of hundreds of units produced. Use the x graph of this function to find the minimum average cost. a) $54 b) $55 c) $100 d) $200 e) $400 f ) None of the preceding. 9. Assume the growth of the population of Del Webb’s Sun City Hilton Head community northwest of Phoenix was linear from 1996 to 2000, with a population of 196 in 1996 and a rate of growth of 705 per year. Identify the equation for the population P of this community as a function of the number of years x after 1996. a) P (x) = 196x + 705 b) P (x) = 196x + 1996 c) P (x) = 1996x + 705 d) P (x) = 705x + 196 e) P (x) = 705x + 1996 f ) None of the preceding. 10. The function that converts euros to U.S. dollars according to Oct. 1, 2008, values is f ( x ) = 1.3816x, where x is the number of euros and f ( x ) is the number of U.S. dollars. (a) Find f -1( x ), the inverse of f . (b) Interpret its meaning. 4 11. A political candidate wishes to use a combination of television and radio advertisements in his campaign. Each 1-minute ad on television reaches 0.18 million eligible voters and each 1-minute ad on radio reaches 0.09 million eligible voters. The candidate feels that he must reach at least 6.3 million eligible voters, but that he should use no more that 60 minutes of advertisements. (a) Write the inequalities that describes these advertising requirements. (b) Graph the region determined by these inequalities in the context of the application. (Be sure to make the scale for each axis.) 12. The revenue from the sale of a product is given by the function R(x) = 3000x – 0.15x3 dollars. Determine how many units should be sold to give a revenue of at least $150,000. 13. The percentage of Fortune Global 500 firms that actively recruited workers on the Internet from 1998 through 2000 can be modeled by P(x) = 26.5x – 194.5 percent where x is the number of years after 1990. Explain why, or why not, this model should be used to predict the percentage of Fortune Global 500 firms that actively recruit workers on the Internet in 1997. 14. Rewrite the expression 4 ln(2a) – ln b as a single logarithm. f x h f x 15. Find the difference quotient when f ( x ) = 3x2 – 2x + 5. h 5 16. A business plans to use three methods of advertising: newspapers (n), radio (r), and cable TV (c) in each of its Table I p r c two markets, Allentown (A) and Brownsville (B). The A 16 12 8 cost per ad type in each market (in thousands of dollars) B 12 10 9 is given in Table I. The business has three target groups: young single females (x), young single males (y), and young married couples (z). Table II gives the number of ads per week directed at each of these groups. Table II p r c x 4 10 12 y 6 14 8 (a) Write a matrix multiplication that shows how the z 8 9 8 advertising money is spent among the target groups in the two markets. (Be sure to label the rows and column in the answer.) (b) Use the result of the multiplication in part (a) to tell which market group has the highest cost, and tell what that cost is. 17. The cost C to obtain drinking water that contains p% impurities for the town of Pleasant View is 85,000 given by the equation C 15,000 . Describe the transformations and/or stretches p 1 needed to obtain this function from the basic reciprocal function C . p 18. If changing market conditions cause a company earning $1,250,000 this year to project decreases of 2% of its profit in each of the next four years, what is the sum of its yearly profits that it projects for the next four years? 6 *** *** Formulas: (1 i ) n 1 kt r S R S = Pe rt S P 1 i k S = P(1 + r) t 1 (1 i) n A R sn a1 1 r n (if r ≠ 1) 1 r i n a1 an sn an = a1 + (n – 1)d an = a1r n – 1 2 7 Answers — Math 1090, FINAL EXAM, Fall 2008 1. sect’n 1.6, chap-rev #63, b 2. sect’n 3.1, #24, b 3. sect’n 4.2, #29, b f ) y = –3.5x2 + 350x + 9732 is good answer 4. sect’n 5.1, chap-rev #39, e 5. sect’n 6.3, #22, d 6. sect’n 3.6, #22, c 7. sect’n 2.2, #59, c 8. sect’n 4.5, #30, d 9. sect’n 1.4, #41, d 10. sect’n 2.7, like # 45, (a) f -1( x ) = 0.7238x (b) one U.S.dollar is worth 0.7238 euros 80 0.18T 0.09 R 6.3 60 11. sect’n 6.1, like #27, (a) T R 60 40 T 0, R 0 20 (b) —> > —> graph —> > —> 12. sect’n 4.6, like #31, 62 ≤ x ≤ 100, between 62 and 100 units, 20 40 60 # television ads 80 inclusive 13. sect’n 1.3, #39, This model should not be used to predict the percentage of Fortune Global 500 firms that actively recruit workers on the Internet in 1997 because the model predict a negative percent which is impossible. 16a 4 b 14. sect’n 3.3, #25, ln 15. sect’n 1.4, like #34, 6x + 3h – 2 4 6 8 x y z 16 12 8 10 14 9 A 280 328 300 T 16. sect’n 5.3, like example #6, (a) [ I ] × [ II ] = 12 10 9 12 8 8 256 284 258 B T or [ II ] × [ I ] = same answer, only transposed (b) The males in Allentown have the highest cost at $328,000 17. sect’n 2.4, like #42, stretches function 85,000 times, and then shifts function up 15,000 units 18. sect’n 64, like #28, $4,754,950.20 8 ┌ * – * –* – * – * – * – * – * – * – * – * – * – * – * – * – * – * – * – * – * – * – * – * – * – * ┐ * Text Sect’n test prob # Text Sect’n test prob # * | | 1.1 - 4.1 - * 1.2 - 4.2 - 3 * | | 1.3 - 13 4.3 - * 1.4 - 9, 15 4.4 - * | | 1.5 - 4.5 - 8 * 1.6 - 1 4.6 - 12 * | | 1.7 - 5.1 - 4 * 1.8 - 5.2 - * | | 2.1 - 5.3 - 16 * 2.2 - 7 5.4 - * | | 2.3 - 6.1 - 11 * 2.4 - 17 6.2 - * | | 2.5 - 6.3 - 5 * 2.6 - 6.4 - 18 * | | 2.7 - 10 * 2.8 - | 3.1 - 2 * 3.2 - | 3.3 - 14 * 3.4 - | 3.5 - * 3.6 - 6 | 3.7 -