Math 1324 Final Review

Transcription

1 Math 134 Final Review 1. (Functions) Determine the domain of the following functions. a) 3 f b) f f c) d) f e) f f) f e g) 1 1 f e h) f ln 5 i) f ln 3 1 j) f ln 1. (1.) Suppose producing tires cost C 5 5,040 and the revenue is R 105 where C ( ) and R ( ) are in dollars. a) What is the profit function? b) What is the break-even quantity? 3. (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 linear, determine how many pencils they must make in a month to break even. 4. (1.) A store can sell 00 units of a certain brand of CD player when the unit price is set at $90. If the prices decreases by $50, the store will be able to sell 100 units. When the unit price is $45, the wholesaler will provide 500 CD players, but if the price increases to $65, the wholesaler will provide an additional 00 units. Find the supply and demand functions (assume they are linear). 5. (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 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 thousands. a) Find the equilibrium quantity. b) Find the equilibrium price. 6. (10.) The profit function in dollars for an sno-cone vender can be given by P , where is the number of sno-cones sold. a) Find the number of sno-cones necessary to maimize profit. b) Find the maimum profit. Fall 017 1

2 Math 134 Final Review 7. (10.) The revenue and cost functions (in dollars) of producing units are given by R 64 and C a) Find the maimum revenue, and the number of units to be sold to obtain that revenue. b) Find the minimum break-even quantity. 8. (10.) A company that produces a certain camera has price-demand function p94.8 5, where p represents the wholesale price per camera at which million cameras can be sold. Also, the company s cost function is C C is the cost in, where millions of dollars for manufacturing and selling million cameras. a) Find the revenue function. b) What is the maimum revenue, and how many cameras must be sold to maimize revenue? c) What camera price will produce maimum revenue? d) For what number of units sold will the company break-even? 9. (.) Solve the system of linear equations: y 35 6y 10. (.) What matri is the result of performing the row operation 5R R3 R on the matri 0 1 7? (.) Solve the system of linear equations: 3z 4 y 3y 3 0 5y 1 3 3z 1. (.) Robert 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 sold eight fewer bottles of water than bottles of iced tea and Gatorade combined. How many bottles of iced tea did he sell? 13. (.) Mr. Parr has been saving his coins. He has a total of 107 coins consisting of nickels, dimes, and quarters. He has nineteen more nickels than he has dimes. If he has $10 altogether, how many quarters does he have? Fall 017

3 Math 134 Final Review 14. (.) The matri below is in reduced row echelon form. Give the solution. If no solution, write no solution. If infinitely many, write in parametric form. w y z (.) Roger wrote and solved a system of equations to find the number of nickels (), dimes (y), and quarters (z) that were in a bank. He found that the general solution was (, y, z) (z 4,8 z, z). List all the solutions to Roger s problem. 16. (.) 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 from the muffins. How many total muffins did they sell that day? 17. (.3) If 3 5d c 5 a c b , then solve for a, b, c, and d. 6 w 3 5 k 18. (.4) Let A 0 m and B p 3 j 8 Give the element that would be in the 3rd row, nd column of the product matri AB. 19. (.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 matri 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? Fall 017 3

4 Math 134 Final Review 0. (.4) An office supply has notebooks in three sizes (1,, and 3 ), and two weights (economy and standard). Table 1 gives the prices for the notebooks. Table gives the quantities of standard notebooks ordered by two businesses. Table 1 Table Standard Standard 3 Standard Economy $1 $ $3 Don s Desks Standard $4 $5 $6 Paul s Pets a) Create two matrices that can be multiplied to determine the amount that each business spent on standard notebooks. b) How much did Paul s Pets spend on standard notebooks? 1. (.4) An epidemic hits a town. Each person in town 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 in matri A and the population of the city, by age and gender, is given in matri B. A Age over 35 Well Sick Carrier B Age Male Female , , , , 000 Over 35 70, , 000 a) Determine if these matrices should be multiplied as AB or BA and find the product matri. b) How many males are sick? c) How many females are well?. (4.1) Create the initial simple tableau for the following Linear Programming Problem. Find the maimum of P subject to: ,, (4.) 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. How many canoes and how many kayaks should he buy in order to maimize the revenue? DEFINE YOUR VARIABLES and SET UP THE PROBLEM. DO NOT SOLVE. Fall 017 4

5 Math 134 Final Review 4. (4.) The matri below is the initial tableau of a linear programming matri. What is the first row and column on which the matri should be pivoted? (4.) The matri below is the final matri from a linear programming problem. 1 3 s1 s P / / 35 4 / / 7 3 / 7 1/ / 7 79 / 7 11/ What is the solution to the problem? Give the values of all variables. 6. (4.) Solve the following Linear Programming Problem using the Simple Method. Maimize P subject to: , (4.) Bob s Burger Barn makes three types of burgers, child, regular, and jumbo. The quantities of ingredients are listed in the chart below. Type: Ground Beef (lbs) Cheese (slices) Pickles (slices) Child 1/10 1 Regular 1/4 1 4 Jumbo /3 6 Suppose Bob s Burger Barn has 800 lbs of ground beef, 4500 slices of cheese, and 1000 pickles available to make burgers. If Bob makes $1.5 profit on each child burger sold, $ profit on each regular burger sold, and $.50 profit on each jumbo burger sold, then determine how many of each burger Bob should sell to maimize his profit. DEFINE YOUR VARIABLES and SET UP THE PROBLEM. DO NOT SOLVE. Fall 017 5

6 Math 134 Final Review 8. (5.1) Jessie invests $8,000 in an account earning 13.5% compounded monthly, and leaves it for 8 years. Determine the effective rate of interest on this account. 9. (5.1) How long, in years, will it take an initial deposit of $5,000 into an account that earns 4.75% compounded monthly to grow to $10,000, if no other funds are deposited, and no withdrawals are made during the term? 30. (5.1) Suppose Ricky deposits $1,750 in a CD that earns 4.5% compounded monthly for 5 years. How much interest does Ricky earn on the account? 31. (5.1) Fred deposited some money into an account that earns 7.5% compounded quarterly. He makes no other deposits or withdrawals, and after 5 years, the accumulated value of the account is $3, How much did Fred deposit? 3. (5.1) James deposits $3,760 in an account where the interest is compounded semi-annually. If he makes no deposits or withdrawals to the account, and at the end of 6 years has $5,80 accumulated in the account, what was the interest rate on the account? 33. (5.) Mark decides to start an annuity. He deposits $15 each week into an account that is compounded weekly at 1.5% interest rate. (a) How much will Mark accumulate in the account if he continues this for 15 years? (b) How much interest will Mark have earned at the end of the 15 years? 34. (5.) Tamara, who is currently 0 years old, decides she wants to have $500,000 by the time she retires at age 65. If she opens an account today that earns.45% interest compounded biweekly, how much money would she need to put in the account every two weeks in order to reach her goal? 35. (5.3) Cheryl owes $300 on her credit card. It is being charged interest at a rate of % compounded monthly. If Cheryl can only afford to pay the minimum required payment of $60 each month, how many years will it take her to pay off this credit card bill? 36. (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 net 8 years? 37. (5.3) Nine years ago, Connor and Jennifer bought a house for $50,000. They paid 30% down and financed the remaining balance at 3.75% annual interest rate compounded monthly for 30 years. a) What are their monthly payments? b) How much will they still owe after paying that monthly payment for 10 years? 38. (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? Fall 017 6

7 Math 134 Final Review 39. (5.3) The Wilsons pay $1,114 each month to pay off their home loan. If the interest on the loan is 7.3% compounded monthly, and the loan will be paid off in 30 years, how much did they borrow? 40. (7.1) Joseph was told to select one of three subjects: algebra (A), biology (B), or chemistry (C)), and to also select one project format (report (R), eperiment (E), poster (P), or tutorial (T)). How many outcomes are in the sample space for this eperiment? 41. (7.1) Jerry tossed a coin, then spun a two-color spinner with red (R) and green (G) sections, and then tossed a 4-sided die (with numbers 1,, 3, and 4 on the faces). Give the sample space for this eperiment. 4. (7.) 75 students were surveyed this morning, 43 said they ate breakfast, 30 said they ate breakfast and read the news, and 5 said they didn t do either of these things. How many students surveyed read the news? 43. (7.) A math professor suggests to her class of 34 students that they should decorate their graphing calculator covers. Four students did not decorate their covers at all. Seventeen students put stickers on the cover. Twenty-one students used paint pens on the cover. How many students used paint pens and put stickers on the covers? 44. (7.) The Venn diagram below shows the results of a survey of 100 people about the pets they owned. a) How many people owned a bird and a cat, but not a dog? b) How many people owned a dog, but not a bird? c) How many people owned a bird or a dog, but not a cat? U Dog Cat 5 16 Bird 45. (7./7.3) The probability Venn diagram below reflects the results of a survey of people regarding the types of power tools they owned. U Drill Saw.13.1 Sander a) What is the probability that a randomly chosen person does not own a drill or a saw? b) What is the probability that a randomly chosen person owns eactly one of the items? c) If a person that owns a saw was randomly selected, what is the probability that he also owns a drill? Fall 017 7

8 Math 134 Final Review 46. (7.) At a party some of the guests brought cookies, drinks, or candy. 5 guests brought cookies, candy and drinks 17 guests brought eactly items 3 guests brought drinks 7 guests did not bring cookies guests brought only candy 15 guests brought cookies, but not drinks 15 guests brought drinks, but not cookies 11 guests brought drinks and candy a) How many total guests were at the party? b) How many guest brought cookies or candy, but not drinks? 47. (7.) 150 A&M students were asked which football activities they attended last weekend. 73 tailgated participated in all three activities 48 did not attend the football game 30 went to midnight yell and the game 47 tailgated, but did not go to midnight yell 19 did not go to any of the three activities 3 tailgated and went to the game, but did not go to midnight yell. U Game Tailgate Midnight Yell a) How many students went to the game, but did not tailgate or go to midnight yell? b) How many students tailgated or went to the game? 48. (7.1/3/4) Two fair si-sided dice are cast and the numbers on top are observed. a) Give event E that at least one of the dice rolled a 4 and the sum is greater than 7. b) Find P(E). c) Find the odds in favor of rolling a sum of 7 or 11. d) Find the probability of not rolling a sum of 7 or (7.3/4) Let E and F be two events in sample space S. Suppose P( E F) 0.31, P( E F) 0.9, and PE ( ) Find: a) P( E F) b) PF ( ) P E F c) 50. (7.3/7.4) One card is drawn from a standard 5-card deck. Find the probability that a five or a club is drawn. Fall 017 8

9 Math 134 Final Review 51. (7.4) In a group of 35 books, 10 are student math books, 4 are teacher math books, 6 are student English books, 8 are teacher English books, and 7 are student physics books. If a book is randomly chosen, what is the probability that it is a teacher book or English book? 5. (7.5) In a council election, the percentage of males and females who voted for the two candidates are shown in the table. Voted for Colton Voted for Stacey Males 9% 31% Females 18% % a) Find the probability that a randomly selected voter is male, given that the person voted for Stacey. b) Find the probability that a randomly selected voter did not vote for Colton. c) Find the probability that a randomly selected female voter voted for Stacey. 53. (7.5) Two fair si-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? 54. (7.3/8.5) 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) According to the data, what is the average number of questions missed on the test? 55. (7.3/8.5) 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 epect somebody from this classroom to have? b) What is the probability a person in the class had received more than tickets? 56. (8.5) A bag contains 3 black marbles, 1 blue marble, 6 red marbles, and 9 green marbles. A game consists of randomly pulling out one marble. If you pull out a black marble, you win $4. If you pull out the blue, you win $8. 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 eperiment. b) Find the epected value of X and eplain what it means. Fall 017 9

10 Math 134 Final Review 57. (8.5) A raffle offers a first prize of $1000, second prizes of $500 each, and 0 third prizes of $10 each. If 10,000 tickets are sold at $1.00 each, find the epected value of X where X denotes the net winnings. Eplain your answer. 58. (10.3) A cost benefit function is given by 7.8 y 100 dollars of removing percent of a given pollutant. where y is the cost in thousands of a) Find the cost of removing 50% of the pollutant. b) Find the cost of removing 80% of the pollutant. c) What percentage of the pollutant can be removed with a budget of $89,700?, where 1 15, 3 C( ) 0 360, (10.3) A cost function is given by and C() is the cost in hundreds of dollars for handling hundred cases of product per month. Find the average cost function. What is the minimum average cost per case (to the nearest dollar)? 60. (10.4) Solve the following eponential equations. Answer with eact answers, not decimal approimations. 4 5 a) e e 0 b) c) e d) e e) e (10.4) Since January of 1980, the growth in a certain population (in hundreds) closely fits A( t) 50e 0. 0t the function defined by, where t is the number of years since a) What was the population in 1995 (to the nearest whole person)? b) In what year did the population hit 7,500? 6. (10.5) Solve the following logarithmic equations. Answer with eact answers, not decimal approimations. ln 38 ln 6 a) b) log c) ln 7 3 d) ln (10.5) The number of years N(r) since two independent evolving languages split off from a common ancestral language is approimated by N( r) 5000ln r, where r is the proportion of the words from the ancestral language common to both languages now. a) If 70% of the words are currently common to languages A and B, how many years have elapsed since the split? (give answer to nearest whole year) b) If two languages split off from a common ancestral language about 1000 years ago, find the proportion of the words common to the languages today. (give to nearest whole percent) Fall