Math 1324 Final Review

Transcription

1 1. (Functions) Determine the domain of the following functions. 3 (a) f x 4x 5x 7 (b) x 4 f x x 1 x 7 f x x x1 f x x (c) (d) 3 1 f x x 5x 6 (e) 5x (f) f x e (g) f x ln 5 x (h) f x ln x 1. (1.) The quantity demanded for a certain brand of CD players is 00 units when the unit price is set at $90. The quantity demanded is 100 units when the unit price is $40. Find the demand equation. 3. (1.) Suppose that the demand and price for potato chips are related by p Dq q where p is the price in dollars and q is the quantity demanded in tens of thousands. Also, suppose the price and supply of the potato chips are related by p S q 0.075q where p is the price in dollars and q is the quantity supplied in tens of thousands. (a) Find the equilibrium quantity. (b) Find the equilibrium price. 4. (1.) Suppose producing x tires cost C x 5x 5,040 and the revenue is Rx 105 where C x and R x are in dollars. (a) What is the profit function? (b) What is the break-even quantity? 5. (10.) The financial department in a company that produces an automatic camera arrived at the following price-demand function and cost function: p x, C x x, where p represents the wholesale price per camera at which x million cameras can be sold. Also, C x is the cost (in millions of dollars) for manufacturing and selling x million cameras. (a) Find the Revenue Function. (b) How many cameras are needed to maximize the Revenue? (c) What is the maximum Revenue? (d) Find the Profit function. (e) How many cameras are needed to maximize the Profit? x Spring 015 1

2 (f) What is the maximum Profit? (g) What should the wholesale price of the camera be in order to maximize Profit? (h) What are the break even points? 6. (10.) The Poage Inflatable Raft Company manufactures inflatable rafts and finds the price x function for the rafts to be px 150 where x represents the number of rafts 50 produced and sold and p(x) is the price of the rafts. Furthermore, the fixed and variable costs to produce x rafts are $5600 and $65 per raft, respectively. (a) What is the Profit function? (b) How many rafts are needed to maximize the Profit? (c) What is the maximum Profit? 7. (10.) Find the vertex and axis of symmetry for f x x 4x (10.) The profit function for a hot dog vender can be given by Px Determine the following. (a) Maximum profit. (b) Number of hot dogs necessary to maximize profit. x 35x (10.) The revenue and cost functions of producing units are given by C x 4x 50. Determine the following. (a) Minimum break-even quantity. (b) Maximum revenue. (c) Maximum profit. (d) Number of units to maximize profit. R x x 64x and 10. (10.) Suppose the demand for a certain item is given by p0 x, where p is the price in 5 dollars when x items are demanded. Also, suppose the cost for selling x items can be modeled by the function C x 4x 70. (a) Find the Revenue function and sketch a graph of it and the Cost function. (b) Determine the break-even point(s). (c) Determine the maximum revenue. (d) What is the price that will produce the maximum revenue? (e) Determine the maximum profit. 11. (10.4) Solve the following exponential equations. Answer with exact answers, not decimal approximations. 4x x 5 (a) e e (b) 3 x 7 6x (c) e 10 Spring 015

3 1. (10.5) Solve the following logarithmic equations. Answer with exact answers, not decimal approximations. ln 3x8 ln 6x (a) (b) log 3x 1 4 (c) ln 7x (.1/) Tanner, Mason, and Cooper went Trick-or-Treating together. Tanner got twice as much candy as Mason. The amount of candy Tanner got was half of what Mason and Cooper got combined. Altogether they received 648 pieces of candy. How many pieces of candy did Mason get? 14. (.1/) Solve the system of linear equations: xy 3x5 6y 15. (.1/) Robert decides to set up a drink stand outside the Academic Building where he sells bottles of water, iced tea, and Gatorade. He sells each bottle of water for $1.5, each bottle of iced tea for $1.75, and each bottle of Gatorade for $.50. At the end of the first day, he has revenue of $48 from the sale of 15 total bottles. He knows that he sold three times as many water bottles as Gatorade bottles. How many bottles of iced tea did he sell? 16. (.1/) Solve the system of linear equations: 3z 4 x y x 3y 3 0 5y 1 3x 3z 17. (.1/) It costs a company $3,500 to produce 3,000 pencils whereas it costs the company $1,500 a month even if they don't produce any pencils. If they sell the pencils for $0.75 and the cost, revenue, and profit functions are known to be linear, determine how many pencils they must make in a month to break even. 18. (.1/) Carr Cafe sells two kinds of muffins. The pumpkin spice muffins sell for $3/each and the banana nut muffins sell for $ each. Last Wednesday they sold twice as many banana nut muffins as they did pumpkin spice muffins, and they brought in a total of $11 just from these muffins. How many total muffins did they sell that day? 19. (.1/) Mr. Parr has been saving his coins. He has a total of 107 coins consisting of nickels, dimes, and quarters. He has four times as many nickels as he does quarters. If he has $10 altogether, how many quarters does he have? Spring 015 3

4 0. (.) The matrix below is in Reduced Row Echelon Form. If the columns, from left to right, are x, y, z, and w read off the answer. If no solution, write no solution. If infinitely many, write in parametric form and give at least examples A (.) Use row operations to turn the -5 (in row 3, column ) into a 0. Which of the following represents the matrix following this row operation? a) b) c) d) e) none of these. (.3) Solve for a, b, c, and d for the following statement. 3 6 a d b 1 7 3c k 7 3. (.4) Let A 0 m 5 8 and B. When you multiply AB, which element 3 4 j 3 is in the 1st row, nd column of the product matrix? Spring 015 4

5 6 w 3 5 k 4. (.4) Let A 0 m and B. When you multiply AB (if possible), p 3 j 8 which element is in the 3rd row, nd column of the product matrix? 5. (.4) An epidemic hits a College Station, TX. Each person is classified by the Health Department as either well, sick, or a carrier. The proportion of people in each category, by age groups, is given by matrix A and the population of the city, by age and gender, is given by matrix B. Age A over 35 Well Sick Carrier (a) Determine if these matrices should be multiplied as AB or BA and find the product matrix. (b) How many sick males are there? (c) How many females are well? B Age Male Female , , , , 000 Over 35 70, , (.4) An office supply has notebooks in three sizes (1,, and 3 ), and two weights (economy and standard). Table A gives the prices for the notebooks. Table B gives the quantities of economy notebooks ordered by two businesses. Table C gives the quantities of standard notebooks ordered by the two businesses. Economy (E) Standard (S) Table A Table B Table C 1 3 $1 $ $3 $4 $5 $6 Jack s Hardware Paul s Pets 1 E E 3 E Jack s Hardware Paul s Pets (a) Create two matrices that can be multiplied to determine the amount that each business spent on economy notebooks. (b) Create two matrices that can be multiplied to determine the amount that each business spent on standard notebooks. (c) How much did Jack s Hardware spend on economy notebooks? (d) How much did Paul s Pets spend on standard notebooks? (e) Using the matrices found in parts (a) and (b), use matrix addition to determine the total amount spent on notebooks for each business. 1 S S 3 S Spring 015 5

6 7. (.4) The prices for Freddy s Filing Cabinets are: $80 for -drawer cabinets, $10 for 3- drawer cabinets, and $160 for 4-drawer cabinets. The table below gives sales numbers for July. Freddy s Filing Cabinets sales for July -drawer 3-drawer 4-drawer Store Store Consider the matrix product: (a) What will the entries in the product represent? (b) What was Store 1 s total revenue for July? (c) What was Store s total revenue for July? x (.4) The SweatShoppe sells three types of shirts in various sizes and styles. Their current inventory and expected marginal profits are summarized in the table below. Inventory Smalls Mediums Larges Extra Larges Short-Sleeved Tees Long-Sleeved Tees Fleece Hoodies Expected Marginal Profits Smalls Mediums Larges Extra Larges Plain $1/item $/item $3/item $4/item Embroidered $/item $3/item $4/item $5/item Create a 3 4 matrix I summarizing the inventory, along with a 4 matrix P representing the marginal profits. (Make sure the information is ordered the same way in your matrices as it is in the given tables.) Then, if possible, multiply the two matrices. (a) Explain the meaning of the entry in the 1 st row, nd column of the product matrix. (b) Explain the meaning of the entry in the nd row, 1 st column of the product matrix. Spring 015 6

7 9. (3.) Solve the following Linear Programming Problem using the Method of Corners. Find the minimum and maximum of P 3x y subject to: 5xy18 1 y 3 x 0 x 8 0 y (3.) Solve the following Linear Programming Problem using the Method of Corners. Find the minimum and maximum of: C 0.18x 0.1 y subject to: x6y 30 4xy 0 y x (3.) Solve the following Linear Programming Problem using the Method of Corners. Find the minimum and maximum of: P x y subject to: yx1 4y x 1 0 y 0 x (3.) The inequalities for a linear programming problem are graphed below. Suppose the point 5,5 is TRUE for all four inequalities. Find the minimum and maximum for the objective function: P x 5y. Spring 015 7

8 33. (3.3) Ryan plans to start a new business called River Run, which will rent canoes and kayaks to people to travel down the Brazos River. He has $5,000 to purchase new boats. He can buy the canoes for $45 each and the kayaks for $700 each. His facility can hold up to 80 boats. The canoes will rent for $0/day and the kayaks will rent for $40/day. If he wants at least three times as many kayaks as canoes, how many canoes and how many kayaks should he buy in order to maximize the revenue? DEFINE YOUR VARIABLES and JUST SET UP THE LINEAR PROGRAMMING PROBLEM. DO NOT SOLVE. 34. (3.3) Bob wants to begin taking vitamin pills to supplement his diet. He decides to only take two types of pills, Pill A and Pill B, for one month. Bob needs at least 9,000 mg of Vitamin A and 18,000 mg of calcium in the month from the two pills. Pill A contains 00 mg of Vitamin A and 100 mg of calcium and Pill B contains 00 mg of Vitamin A and 300 mg of calcium. Bob would like to take at least twice as many of Pill B than Pill A. If each pill costs $0.0 per Pill A and $0.30 per Pill B, how many of each pill should Bob buy to minimize his cost? What is the minimum cost for the pills? 35. (5.1) Robert deposits $30 into an account earning simple interest. What is the interest rate if ten months later he has $450 in the account. 36. (5.1) Determine the effective rate of an account that earns 13.5% compounded monthly. 37. (5.1) How long, in years, will it take an account that earns 4.75% compounded monthly to double in value if nothing is deposited into or out of the account during the term? 38. (5.1) Suppose Sam deposits $1,750 in a CD that earns 4.5% compounded monthly for 5 years. How much will be in the account after the 5 years? 39. (5.1) Suppose Fred deposits $5,000 into an account that earns 7.5% compounded quarterly. If he makes no deposits into or out of the account over its term, then how long will it take (in years) for his account to reach $6,500? 40. (5.) Mark decides to start an annuity. He deposits $15/week into an account that is compounded weekly at 1.5% interest rate. How much interest would Mark earn on this annuity if he continues this for 15 years? 41. (5.) Tamara, who is currently 0 years old, decides she wants to have million dollars by the time she retires at age 65. If she opens an account today that earns.45% interest compounded daily, how much money would she need to put in the account each day in order to reach her goal? 4. (5.3) Rico figures he can afford a car payment of $300/month if the interest rates remain at 4.9%/year compounded monthly. If he plans to put $4000 down and finance the rest for 6 years, what is the most the original price (cash price) of the car can be? Spring 015 8

9 43. (5.3) Cheryl racked up a credit card bill of $300. It is being charged interest at a rate of % compounded monthly. If Cheryl can only afford to pay the minimum required payment each month, of $60. How many years will it take her to pay off this credit card bill? 44. (5.3) How much money should Jordan deposit into an account today, at.9% interest compounded quarterly, so that it will yield payments of $1000 at the end of each quarter for the next 8 years? 45. (5.3) Nine years ago, Connor and Jennifer bought a house for $75,000. They paid 30% down and financed the remaining balance at 3.75% annual interest rate compounded monthly for 30 years. What are their monthly payments? 46. (5.3) Reece and Sandra bought a house. They financed it for $185,000 at 5.7% interest compounded monthly for 30 years. (a) What are their monthly payments? (b) How much of their very first payment went towards the principle of the loan? (c) How much of their very first payment went towards the interest of the loan? (d) If they paid $100 more than the minimum monthly payment, how many years will it take to pay off the loan? 47. (5.3) Jennie and Marshall buy a house for $16,500. They have no money to put as a down payment, so they finance the whole thing at 7.3%/year interest rate compounded monthly for 30 years. What are their monthly payments? 48. (7.1) For any Universal set, with subsets A, B, and C, CLEARLY circle True or False for each of the following. True/False if A B, then A and B True/False A A True/False A B A B True/False if A C and C A then A C True/False C C U 49. (7.1) Let U = {1,, 5, 7, 9, 10, 11, 1, 14}, A = {1, 5, 9, 1}, B = {, 7, 1, 14}, C = {1, 10, 11}, and D = {5, 9}. True/False A A A D True/False True/False D A B True/False n A B 4 True/False A and B are disjoint True/False D D Spring 015 9

10 50. (7.) If nu 33, n A B 9, n A B 5, and nb 3, find 51. (7.) At a Halloween party some of the guests brought cookies, drinks, or candy. How many total guests were at the party? 5 guest brought cookies, candy and drinks 19 guests brought exactly items 1 guests brought drinks 0 guests did not bring cookies 8 guests brought only candy guests did not bring a drink or candy 5 guests brought only cookies and drinks 11 guests brought drinks and candy n A. 5. (7.) 75 students were surveyed this morning, 59 said they ate breakfast, 48 said they ate breakfast AND read the Battalion, and 7 said they didn t do either of these things. How many students surveyed read the Battalion? 53. (7.) A math prof suggests to her class of 34 students that they should decorate their graphing calculator covers. 4 students end up not decorating them at all. 17 students put stickers on the cover. 1 students used paint pens on the cover. How many students put stickers on and used paint pens? 54. (7.) The Venn diagram to the right shows the number of elements in each space. Use this Venn diagram to answer the following two questions. (a) Find (b) Find n AB C. n B C A. Spring

11 55. (7.) A local high school math teacher conducted a survey to find out the correlation between students who were suspended from school and whether or not they passed their math class. How many girls were suspended and failed? NOTE: Two of the numbers in the Venn diagram have been filled in for you. 51 boys were surveyed 10 boys who got suspended still passed 4 students failed 48 girls did not get suspended 30 boys passed the class 17 boys were suspended and failed 56. (7.) A survey of 150 A&M students were asked which football activities they attended last weekend. Use the Venn diagram below to help figure out how many students ONLY went to the game. (note: numbers have been filled in for you) 73 tailgated went to midnight yell, tailgated, and attended the football game 48 did not attend the football game 30 went to midnight yell and the game 47 tailgated, but did not go to midnight yell 57. (7.) Let nu 6, n A B 6, n A 14, and nb 11. Find n A. 58. (7.3/4) Two fair six-sided dice are cast and the numbers on top are observed. (a) What is the event that at least one of the dice rolled a 4 and the sum of the dice is greater than 7? (b) Find the odds in favor of rolling a sum of 7 or (7.3/4) Let E and F be two events in the same sample space. Suppose PE 0.9, PE 0.31, and PE 0.45 (a) PF (b) PE F (c) PE F F. Find F Spring

12 60. (7.4) Among 00 students at Blinn, 87 said they are currently taking a math course. Sixtythree said they are currently taking an English class. 91 students are not taking a math or English class. What is the probability a student is currently taking English, but not math? 61. (7.4) A survey was conducted of 475 students living in dorms at Texas A&M. There were 37 students who said they had a refrigerator in their dorm room, 186 said they had a microwave in their dorm room, and 30 who said they did not have a refrigerator nor a microwave in their dorm room. What is the probability a student selected at random had BOTH a refrigerator and a microwave in their dorm room? 6. (7.4) One-hundred-fifty people were asked what they put in their coffee. 54 said they used cream, 7 said they used sugar. 6 said they did not use cream or sugar. What is the probability a person selected from this group used cream and sugar? 63. (7.4) Let E and F be two events of the same sample space. If PE PF PE F 0.5, what is PE F? 0.55, 0.70, and 64. (7.5) Two fair six-sided dice are cast (one green and one red). What is the probability the red die rolled an even number, if it is known that the sum of the two dice was greater than 8? 65. (7.5) Three cards are drawn in succession without replacement. What is the probability the 1st card is a diamond, given that the nd card is a club and the 3rd card is a diamond? 66. (7.5/6) An experiment consists of randomly selecting one of three coins, tossing it, and observing the outcome - heads or tails. The first coin is a fair coin, the second coin is a two headed coin, and the third coin is a biased coin such that P (H) = (a) What is the probability the coin lands on tails? (b) Given that the coin landed on heads, what is the probability it was the fair coin? 67. (7.5/6) In the 008 presidential election, over 130 million people voted, of which 46.3% were male. Of the male voters, 49% voted for Obama and 48% voted for McCain. Of the female voters, 56% voted for Obama and 43% voted for McCain. Source: U.S. Census and The New York Times (a) Find the probability that a randomly selected voter is a male, given that he voted for McCain. (b) Find the probability that a randomly selected voter did not vote for McCain or Obama (neither one). 68. (7.6) Box A has green and 5 blue marbles in it. Box B has 9 green and 6 blue marbles in it. A marble is drawn from Box A, transferred to Box B, and then a marble is drawn from Box B. What is the probability a blue marble was drawn from Box A, if you know a green marble was drawn from Box B? 69. (7.6) In a certain area, 18% of the population are joggers and 45% of the joggers are women. If 60% of those who do not jog are women, find the probability that a person selected at random from this community is a jogger given that the person is a man? Spring 015 1

13 70. (7.5/6) Using the tree diagram below, answer the following 5 questions. (a) Find PD. (b) Find PB E. (c) Find PC D (d) Find P A E.. (e) Find PE C. 71. (7.6) A study conducted in America found that 5% of Stay-At-Home Moms maintain a blog. It also found that 8% of Working Moms blog and 4% of women who aren t Moms blog. If the women of America consist of 4% Stay-At-Home Moms, 67% Working Moms, and the rest are not Moms, find the probability that a person who blogs is a Stay-At-Home Mom. 7 (8.1) Tom has 3 different CDs of Elvis, different CDs of the Black Eyed Peas and 4 different CDs of Aerosmith. In how many ways can those CDs be placed on a shelf so that CDs by the same group are next to each other? 73. (8.1) Certain codes are formed with letters, followed by 3 digits, followed by more letters (no spaces). If the first letter must be a vowel, the first digit must be odd, and no letter or digit can be repeated, how many different codes are possible? 74. (8.1) Sandra, Katy, and Ashley go to the movies with 4 other friends. In how many ways can this be done if either Sandra, Katy, or Ashley MUST sit in the middle seat? 75. (8.1) A group of FRIENDS (Monica, Chandler, Phoebe, Ross, Rachel, & Joey) go to a coffee shop. They all want to sit on the large couch side-by-side. In how many ways could they sit down if Ross MUST sit by Rachel? 76. (8.1) In how many ways can 8 graduate students and 5 undergraduate students be seated in a row if graduates sit on one side and undergraduates sit on the other side? 77. (8.1) A group consists of 3 boys and 6 girls. They are all arranged randomly in a row. What is the probability a girl is at each end and a girl is in the middle seat? 78. (8.) A band has 9 officers consisting of a Drum Major, an Assistant Drum Major, a Band Sweetheart, and 6 Council Members. There are 193 boys among the 354 band members. If the Drum Major must be a boy and the Band Sweetheart must be a girl, in how many different ways could these officers be selected? (SHOW ALL WORK!) 79. (8.) A box contains 75 light bulbs, of which 8 are defective. If Matthew randomly selects 10 light bulbs, in how many ways could he select at least 1 defective light bulb? Spring

14 80. (8.) Mrs. Teacher is selecting a group of students to help her grade the exams. She needs a coordinator, a grade recorder, 3 graders for the multiple choice, and 5 graders for the workout problems. The coordinator and grade recorder must be selected from 7 graduate students. The graders must be selected from 13 undergraduate students. (NOTE: Each student can only hold one position and graders cannot grade both multiple choice and work-out problems.) In how many ways can Mrs. Teacher select her helpers? 81. (8.) Katniss randomly has to select 6 markers out of a box containing 8 black, red, 6 blue, and 3 green markers. In how many ways is this possible if she must select exactly 4 of the same color? 8. (8.3) Cheryl has a cooler with 4 Dr. Peppers and 7 Cokes inside. If drinks are randomly drawn out, in succession, without replacement, what is the probability that the two drinks are the same brand? 83. (8.3) A box of jawbreakers contains 4 blue, red, 6 green, 1 purple, and yellow jawbreakers. If Chris randomly selects 5 jawbreakers, what is the probability he gets exactly 3 the same color? 84. (8.3) Four boys and five girls are randomly assigned seats in a row. What is the probability that all the girls sit together and all the boys sit together? 85. (8.5) A bag contains 3 black bean-bags, 1 blue bean-bag, 6 red bean-bags, and 9 green beanbags. A game consists of randomly pulling out bean-bags at the same time. If you pull out two of the same color, you win $4. If you pull out the blue, you win $6. For everything else, you lose your money. It costs $1.50 to play the game. Let the random variable X denote the net winnings. (a) Find the probability distribution associated with this experiment. (b) Find the expected value of X and explain your answer. 86. (8.5) A box of t-shirts has 7 small, 3 medium, 5 large, and 4 X-large t-shirts inside. Eric pulls out t-shirts 1 at a time until he has small. Let the random variable, X, denote the number of shirts he pulls out of the box. What values may X assume? 87. (8.5) Two fair six-sided dice are rolled at the same time. Let the random variable, X, denote the sum of the two dice. Find the probability distribution for X. 88. (8.5) A raffle offers a first prize of $1000, second prizes of $300 each, and 0 third prizes of $10 each. If 10,000 tickets are sold at 50 cents each, find the expected value of X where X denotes the net winnings. Explain your answer. Spring

15 89. (8.5/9.) A survey was conducted among a group of students to see how many questions they missed on their last history test. Below is a table of the results. # of questions missed frequency (a) What is the probability a student missed less than five questions? (b) What is the variance for the number of questions missed? (c) According to the above data, how many questions could a student expect to have missed on this test? 90. (8.5/9.1/) A classroom of people were asked how many speeding tickets they had received during their lifetime. Let the random variable, X, denote the number of speeding tickets. Based upon these results, answer the following 4 questions. number of people number of speeding tickets (a) How many tickets would you expect somebody from this classroom to have? (b) What is the median for the number of tickets these people had received? (c) What is the probability a person in the class had received more than tickets? (d) What is the standard deviation in the number of tickets received? 91. (8.5/9.1/) A survey was conducted among a group of students to see how many questions they missed on their last history test. Below is a table of the results. # of questions missed frequency (a) According to the above data, how many questions could a student expect to have missed on this test? (b) What is the variance for the number of questions missed? (c) What was the mode? (d) What was the median? 9. (8.5/9.1/) A bar is missing above the 5 to complete the histogram to the right. Determine how high the bar above the 5 should be drawn and answer the five questions below. (a) What is the mode? (b) What is the standard deviation? (c) What is the median? (d) What is the variance? (e) What is the mean? Spring

16 93. (9.1/) The following are the final exam scores for a math class. 45, 65, 67, 67, 75, 75, 75, 83, 89, 90, 90, 90, 90, 94, 96, 99, 100, 100 (a) What is the mean exam score? (b) What is the median exam score? (c) What is the mode? (d) What is the standard deviation for the exam scores? (e) What percentage of the exam scores are within one standard deviation from the mean? 94. (9.3) The average GPA for the graduating class at Bryan High School this year was a 1.96 with a standard deviation of What is the lowest GPA a student in the graduating class could have and still be in the top 5% of the class? 95. (9.3) Flush-It-Out Inc. manufactures pipes. These pipes are accepted by the buyer if they fall within the tolerance limits of 0.79 and 0.87 inches in length. Assuming that the length of these pipes is normally distributed with a mean of 0.8 inches and a standard deviation of 0.05 inches, estimate the percentage of pipes that will be rejected by the buyer. 96. (9.3) The mean clotting time of blood is 7.45 seconds with a standard deviation of 3.6 seconds. What is the probability that an individual s clotting time will be less than 3 seconds or greater than 13 seconds? Assume normal distribution. 97. (9.3) Let Z be the Standard Normal Random Variable. Find a, such that P ( a Z a) = (9.3) The life span of a 60 watt light bulb is normally distributed with an average life span of 8,000 hours and a standard deviation of 15 days. What is the probability that a bulb selected at random will last at least 8,50 hours? Spring