Finite Math Departmental Review Fall 2014

Transcription

1 Finite Math Departmental Review Fall 2014 The only materials allowed for use during the final exam will be a pen or pencil, a TI 82, 83 or 84 calculator, a copy of the departmental formula sheet, and a z-table. All work must be shown for full credit. 1. Find the simple interest on $31, 400 at 6.25% for 193 days. Assume 360 days in a year and 30 days in a month A student borrows $4000 at 10% for 9 months to pay tuition. Find the total amount due An account invested in a money market fund grew from $73, to $73, in a month. What is the interest rate to the nearest tenth? Dave took out a $6500 loan at 9% and eventually repaid $8255 (principal and interest). What was the time period of the loan? A newborn child receives an $8000 gift toward a college education from her grandparents. How much will the $8000 be worth in 19 years if it is invested at 7.2% compounded quarterly? Ben Franklin s gift of $10,000 to New York City grew to $6,000,000 in 200 years. At what interest rate compounded annually would this growth occur? Bank One offered an 18 year certificate of deposit (CD) at 4.23% interest compounded quarterly. ON the same day on the internet, First Bank offered an 18 year CD at 4.22% compounded monthly. Find the APY for each CD. Which bank paid a higher APY? A company will need $55,000 in 8 years for a new addition. To meet this goal, the company deposits money in an account today that pays 9% annual interest compounded quarterly. Find the amount that should be invested to total $55,000 in 8 years If inflation has been running at 2.75% per year and a new car costs $27,100 today, what would it have cost 5 years ago? At the average annual inflation rate of 4.4%, about how long would it take for the general level of prices in the economy to double?

2 11. In order to accumulate enough money for a down payment on a house, a couple deposits $256 per month into an account paying 3% compounded monthly. If payments are made at the end of each period, how much money will be in the account in 4 years? A full-time worker aged 30 invests $250 a month in a fund which has an average yearly return of 6.0% compounded monthly. A) How much money will the worker have in his fund when he is 60 years old? B) If the worker makes no further deposits and makes no withdrawals after age 60, how much will he have for retirement at age 68? An investor needs $21,000 in 14 years. A) What amount should be deposited in a fund at the end of each quarter at 9% compounded quarterly so that there will be enough money in the fund? B) Find the investor s quarterly deposit if the money is deposited at 6.5% compounded quarterly Ingrid wants to buy a $15,000 car in 8 years. How much money must she deposit at the end of each quarter in an account paying 5.8% compounded quarterly so that she will have enough to pay for her car? Find the lump sum deposited today that will yield the same total amount as payments of $15,000 at the end of each year for 17 years, at an interest rate of 4% compounded annually Fritz Benjamin buys a car costing $15,200. He agrees to make payments at the end of each monthly period for 4 years. He pays 9.6% interest, compounded monthly. What is the amount of each payment? Find the total amount of interest Fritz will pay Susan Carver will purchase a home for $410,000. She will use a down payment of 20% and finance the remaining portion at 8.7%, compounded monthly for 30 years. A) What will be the monthly payment? B) How much will remain on the loan after making payments for 8 years? C) How much interest will be paid on the total amount of the loan over the course of 30 years? A woman, with her employer s matching program, contributes $500 at the end of each month to her retirement account, which earns 9% interest, compounded monthly. When she retires after 42 years, she plans to make monthly withdrawals for 25 years. If her account earns 5% interest, compounded monthly, then when she retires, what is her maximum possible monthly withdrawal (without running out of money)?

3 19. Large semitrailer trucks cost $120,000 each. A trucking company buys such a truck and agrees to pay for it by a loan that will be amortized with 8 semiannual payments at 18% compounded semiannually. Complete an amortization schedule for the first four payments of the loan Payment number Amount of payment Interest payment Applied to principal Balance 20. A college student works in both the school cafeteria and library. She works no more than 15 hours per week at the cafeteria, and no more than 18 hours per week at the library. She must work at least 20 hours per week. Write a system of inequalities that describes all the given conditions. Graph the feasible region Use the graphing method to find the minimum and maximum values of z = 5x + 6y, if possible, for the following set of constraints x + 4y 12 x + 4y 8 x 0, y Use the graphing method to find the minimum and maximum values of z = 9x + 5y, if possible, for the following set of constraints x + y 7 x + y 1 2x y It takes 9 units of carbohydrates and 5 units of protein to satisfy David s minimum weekly requirements. The meat contains 2 units of carbohydrates and 2 units of protein per pound. The cheese contains 3 units of carbohydrates and 1 unit of protein per pound. The meat costs $3.50 per pound and the chess costs $4.40 per pound. How many pounds of each are needed to fulfill the minimum requirements at minimum cost? What is David s minimum cost? Use the graphing method

4 24. Kimo s Material Company hauls gravel to a construction site, using a small truck and a large truck. The carrying capacity and operating cost per load are given in the accompanying table. Kimo must deliver a minimum of 200 cubic yards per day to satisfy her contract with the builder. The union contract requires that the total number of loads per day is a minimum of 6. Use the graphing method to determine how many loads should be made in each truck per day to minimize the total cost Small truck Large truck Capacity (cubic yds) Cost per load $83 $ Solve the following linear programming problem using the simplex method Maximize z = 2x + 5y subject to 5x + y 80 5x + 2y 100 x + y 90 x 0, y Use the simplex method to solve the linear programming problem Maximize: z = 2x + y Subject to: x + 3y 12 2x + 3y 6 x + y 4 with x 0, y Use the simplex method to solve the following: Maximize f = 2x + 2y 4z Subject to 3x + 3y - 6z 57 6x + 6y + 12z 128 x 0, y 0, z 0

5 28. Use the simplex method to solve the following: Maximize f = 4x - 3y + 2z Subject to 2x - y + 8z 120 4x - 5y + 6z 180 2x - 2y + 6z 96 x 0, y 0, z A cat breeder has the following amounts of cat food: 120 units of tuna, 110 units of liver, and 70 units of chicken. To raise a Siamese cat, the breeder must use 2 units of tuna, 2 of liver and 3 of chicken, while raising a Persian cat requires 3, 2, and 1 units respectively per day. If a Siamese cat sells for $14 and a Persian cat sells for $16, how many of each should be raised in order to obtain maximum gross income? Set up the system of equations and initial simplex tableau. Do not solve the problem A bicycle manufacturer builds one, three and ten speed models. The bicycles need both aluminum and steel. The company has available 54,125 units of steel and 43,980 units of aluminum. The one, three and ten speed models need, respectively, 10, 20 and 25 units of steel and 16, 8, and 20 units of aluminum. How many of each type of bicycle should be made in order to maximize profit if the company makes $6 per one speed bide, $12 per three speed bike, and $30 per ten speed bike. What is the maximum possible profit? A company sells sets of kitchen knives. A Basic Set consists of 2 utility knives and 1 chef s knife. A Regular Set consists of 2 utility knives, 1 chef s knife, and 1 slicer. A Deluxe Set consists of 3 utility knives, 1 chef s knife and 1 slicer. The profit is $30 on a Basic Set, $40 on a Regular Set, and $70 on a Deluxe Set. The factory has on hand 1600 utility knives, 800 chef s knives, and 400 slicers. If all sets will be sold, how many of each type should be made up in order to maximize profit? What is the maximum profit? Use the simplex method to solve the following problem: Maximize: z = 12x + 10y Subject to : x + 5y 24 x + y 49 x 0, y 0

6 33. Solve the following problem using the simplex method: Maximize: f = 3x + 2y + 2z Subject to: x + y + 2z 33 2x + y + z 23 x 0, y 0, z Solve using the simplex method: Minimize w = 7x + 6y Subject to: 6x + 7y 420 x + 4y 40 x 0, y Use the simplex method to solve the following problem: Find x 0, y 0, and z 0 such that 5x + 4y + 9z 18 5x + 8y + 2z 10 And w = 25x + 16y + 64z is minimized. 36. Let D = {8,11,13}, E = {8,10,11,12} and F = {7,9,10,11,13} List the elements in the set (D U E) F. 37. Shade a Venn diagram to represent the set U Shade a Venn diagram to represent the set (C A ) U B In 2007, the percentage of children under 18 years of age who lived with both parents was 68.5, the percentage of children who lived only with their mother was 27.3, and the percentage of children who lived with neither parent was 2.8. Use a Venn diagram to determine the percentage of children under age 18 who lived with their father only

7 cities were surveyed to determine sports teams. 24 had baseball, 21 had rugby, 15 had volleyball, 13 had baseball and rugby, 10 had baseball and volleyball, 10 had rugby and volleyball. 6 had all three. Create a Venn diagram and use it to answer the following questions: a) How many had only a baseball team? b) How many had baseball and rugby, but not volleyball? c) How many had baseball or rugby? d) How many had baseball or rugby, but not volleyball? e) How many had exactly two teams? 41. The table lists the cross-classifications of marital status and sex for the adults chosen for the 2006 General Social Survey (GSS). Using the letters given in the table, find the number of respondents in each of the following sets in parts a) through d) Marital Status Male (M) Female (F) Married (A) Widowed (B) Divorced (C) Separated (D) Never Married (E) a) E M b) B ( F U M) c) E U M d) B D 42. A man wants to adopt a puppy from an animal shelter. At the shelter, he finds eight puppies that he likes, a male and female puppy from each of four breeds of terrier, boxer, dachshund, and Labrador. The puppies are so cute that he can t make up his mind, so he decides to the pick the dog randomly. Find the probability that the man chooses anything but a Labrador

8 43. Several friends chartered a boat for a day s fishing. The caught a total of 68 fish. The table below provides information about the type and number of fish caught. Determine the empirical probability that the next fish caught is a kingfish Fish Number caught Grouper 16 Shark 19 Flounder 7 Kingfish When playing American roulette, the croupier (attendant) spins a marble that lands in one of the 38 slots in a revolving turntable. The slots are number 1 to 36, with two additional slots labeled 0 and 00 that are painted green. Consider the numbers 0 and 00 as neither even nor odd. Half the remaining slots are colored red and half are black. Assume a single spin of the roulette wheel is made. Find the probability of the marble landing on a 00 or even slot A pair of dice is rolled. Find the probability of rolling a) a sum not more than 9 b) a sum not less than 5 c) a sum between 6 and 10 (exclusive). 46. The mathematics faculty at a college consists of 5 professors, 6 associate professors, 12 assistant professors, and 12 instructors. If one faculty member is randomly selected, find the probability of choosing a professor or an instructor Let P(Z) = 0.40, P(Y) = 0.42, and P(Z Y) = Use a Venn diagram to find a) P(Z Y ), b) P(Z U Y ), c) P(Z U Y) and d) P(Z Y ) According to a recent survey, the probability that a country s resident earns $75,000 or more a year (event A) is The probability that a country s resident lives in an owner-occupied home (event B) is The probability that a country s resident earns $75,000 or more and lives in an owner-occupied home is Create a Venn diagram and use it to find the following probabilities: a) P(A U B) b) P(A B) c) P(A B ) d) P(A U B )

9 49. A single, fair 6-sided die is tossed. Find the odds in favor of rolling a 3 or a A pair of dice is rolled. What are the odds of rolling a sum of 10 or a sum of 11? The table below is based on numbers from a survey. The table represents 2067 randomly chosen residents of a country 25 years or older in regard to educational attainment and health insurance status. Find the probability of not having a bachelor s degree or not having health insurance Educational attainment Has health insurance Does not have health insurance Less than high school graduate High school graduate (or GED) Some college or an associate s degree Bachelor s degree or higher A bicycle factory runs two assembly lines, A and B. 98% of line A s products pass inspection and 94% of line B s products pass inspection. 50% of the factory s bikes come off assembly line B and the rest come off line A. Find the probability that one of the factory s bikes did not pass inspection and came off assembly line A Suppose for 2011, 39.6% of the civilian population 16 years or older was male, 38% was not in the labor force, and 65.3% was male or not in the labor force. Find the probability of not being in the labor force, given that the person is male The table shows the result of a restaurant survey. Find the probability the service was poor, given that the meal was dinner Meal Service good Service poor total Lunch Dinner Total

10 55. A survey determined annual household income and whether the respondent found the work stressful. The results are in the table below. Given that the respondent sometimes has stress, what is the probability of earning $50,000 or less? Stress at work Income Always Often Sometimes Hardly ever Never totals $50,000 or less $50,001 - $100, More than $100, Total Suppose you have 3 jars with the following contents. Jar 1 has 4 white balls and 1 black ball. Jar 2 has 2 white balls and 3 black balls. Jar 3 has 1 white ball and 3 black balls. One jar is to be selected and then 1 ball is to be drawn from the selected jar. The probabilities of selecting the first, second and third jars are ½, 1/3, and 1/6 respectively. Find the probability the ball was drawn from Jar 2, given that the ball is white Data from the Bureau of Labor Statistics indicates that in a certain month, 30.3% of the labor force had a high school diploma or fewer years of education, 23.6% had some college or an associate s degree and 46.1% had a bachelor s degree or more education. Of those with a high school diploma or fewer years of education, 5.7% were unemployed. Of those with some college or an associate s degree, 4.4% were unemployed, and of those with a bachelor s degree or more education, 4.2% were unemployed. Find the probability that a randomly chosen labor force participant has a high school diploma or less education given that he or she is employed IN 2013, the total sales form 3 different car production companies was 606,334 cars or light trucks. The probability that the vehicle was made by Company A was 0.381, by Company B 0.350, and by Company C Additionally, the probability that a Company A vehicle sold was a car was 0.392, a Company B vehicle sold was a car was 0.370, and a Company C vehicle sold was a car was Given the vehicle sold was a car, find the probability that it was made by Company A For the experiment described, let x determine a random variable, and use your knowledge of probability to prepare a probability distribution. Three balls are drawn (with replacement) from a bag that contains 6 green balls and 3 yellow balls. The number of yellow balls is counted Two names are drawn from a hat, signifying who should go pick up pizza. Five of the names are on the swim team and four are not. The number of swimmers selected is counted. Draw a histogram, and shade the region that gives the probability of at least one swimmer s name being drawn

11 61. Find the expected value of the random variable x P(x) Suppose you are invited to play a game in which you bet $1 on any number from 000 to 299. If your number comes up, you get $75. Find the expected winnings A raffle offers a first prize of $1000, 2 second prizes of $300, and 20 third prizes of $10 each. If 10,000 tickets are sold at 75 cents each, find the expected winnings for a person buying 1 ticket A contest at a fast-food restaurant offered the following cash prizes and probabilities of winning on one visit when a customer buys an order of French fries for $1.80. Find the expected winnings if the player buys 9 orders of French fries in multiple visits Prize probability $100,000 1/8,504,860 $50,000 1/302,500 $10,000 1/282,735 $1,000 1/153,560 $100 1/104,560 $25 1/ An insurance company has written 50 policies for $100,000, 100 policies for $45,000 and 250 policies for $15,000 for people of age 20. If experience show that the probability that a person will die at age 20 is , how much can the company expect to pay out during the year the policies were written? Otitis Media, or middle ear infection, is initially treated with an antibiotic. Researchers have compared two antibiotics, A and B, for their cost effectiveness. A is inexpensive, safe, and effective. B is also safe. However, it is considerable more expensive and it is generally more effective. Use the table (where costs are estimated as the total cost of medication, office visit, ear check, and hours of lost work) to find the expected cost of using each antibiotic. To minimize total expected cost, which antibiotic should be chosen? cure Cost $59.30 Antibiotic A 0.30 no cure Cost $ cure Cost $69.15 Antibiotic B 0.20 no cure Cost $106.00

12 67. Pizza House offers 2 different salads, 5 different kinds of pizza and 4 different desserts. How many different three course meals can be ordered? The owner of a stereo store wants to advertise that he has many different sound systems in stock. The store carries 10 different CD players, 6 different receivers, and 7 different speakers. Assuming a sound system consists of one of each, how many different sound systems can he advertise? How many different 4-letter words can be made a) if the first letter must be B, Y, P, or M and no letter may be repeated? b) if repeats are allowed (but the first letter is B, Y, P, or M)? c) How many of the 4-letter words (starting with B, Y, P, or M) with no repeats end in H? License plate tags in a particular state are to consist of 3 letters followed by 4 digits with repeated letters and digits allowed. How many different license plate tags can there be in this state? For a segment of a radio show, a disc jockey can play 6 records. If there are 10 records to select from, in how many ways can the program for this segment be arranged? Mrs. Welch and her 6 children are shopping at a local grocery store. Each of the children will be allowed to select one box of cereal for their own from the 9 different boxes of cereal available (there are no two the same). In how many ways can the selections be made? In a certain lottery, the player chooses 4 numbers from the numbers 1 through 31. In how many ways can the 4 numbers be chosen? A committee of 8 students is to be selected from 11 students in a fraternity a) In how many ways can this be done? b) In how many ways can the group that will not take part be chosen? 75. How many different committees can be formed from 6 teachers and 44 students if the committee consists of 2 teachers and 4 students? An ice cream store sells 22 flavors of ice cream. Determine the number of possible 6-dip sundaes a) if order is considered and no flavor can be repeated? b) if order is not considered and no flavor can be repeated?

13 77. A container contains 12 diesel engines. The company chooses 5 engines at random and will not ship the container if any of the engines chosen are defective. Find the probability that a container will be shipped even though it contains 2 defectives if the sample size is A shipment of 9 typewriters contains 4 that are defective. Find the probability that a sample of size 3, drawn from the 9, will not contain a defective typewriter A radio station runs a promotion at an auto show with a money box with 10 $50 tickets, 12 $25 tickets and 11 $5 tickets. The box contains an additional 20 dummy tickets with no value. Find the probability that exactly two $50 prizes and one no money winner are chosen when 3 tickets are randomly drawn A bridge hand is made up of 13 cards from a deck of 52. Find the probability that a hand chosen at random contains at least 3 nines A shipment contains 10 igneous, 8 sedimentary, and 8 metamorphic rocks. If 4 rocks are selected at random, find the probability that exactly 2 are sedimentary What is the probability of winning a lottery in which you must choose 6 numbers from the numbers 1 through 13? a) Assuming that order is unimportant b) Assuming that order is important. 83. The sales force of a business consists of 10 men and 10 women. A production unit of 8 people is set up at random. a) What is the probability it will consist of 3 men and 5 women? b) What is the probability it will consist of all men? c) What is the probability it will consist of at least one woman? 84. Twenty-three subjects volunteer for a study of a new cold medicine. Eleven of the volunteers are ages 20-39, 6 are ages 40-59, and 6 are age 60 or older. If 8 volunteers are selected at random, find the probability of each of the following: a) All the volunteers selected are ages b) 5 of the volunteers are ages and 3 are age 60 or older. c) 3 of the volunteers are ages

14 % of the population is 65 or older. Find the probability that the following number of persons selected at random from 50 people are 65 or older a) None are 65 or older. b) At most 2 are 65 or older. 86. A study determined that 6% of children under 18 years old lived with their father only. Find the probability that none of the children selected at random from 15 children under 18 years old lived with their father only A coin is tossed 5 times. Find the probability of getting at least 3 heads A study determined that 7% of the persons age 65 to 74 have a certain symptom. A group of 16 persons age 65 to 74 are selected at random. Find the probability that at least 2 persons have the symptom According to an article, 35% of all cars crossing a toll bridge have a commuter sticker. What is the probability that among 13 randomly selected cars waiting to cross the bridge, at least 2 have commuter stickers? The amounts (in dollars) spent by 20 families for dinner at a restaurant are given below. Construct a frequency table based on the data using 10 dollar intervals. Then construct a histogram that represents the data set Find the mean of the estimated miles per gallon of the automobiles in the table Value frequency For the following set of numbers, find the median , 59, 55, 34, 39, 46

15 93. The grouped frequency distribution for the weight gain (in pounds) for 25 pregnant women is given below. Find the mean of the grouped data and determine which interval is the modal class Interval frequency Find the mean, median, and mode for the daily auto sales at a local car dealership , 12, 13, 15, 16, 26, Find the range and standard deviation for the set of numbers , 130, 138, 148, 152, 160, Find the standard deviation for the grouped data College units frequency

16 97. Use the table, which gives the total amounts of sales (in millions of dollars) for aerobic, basketball, cross-training, and walking shoes for the years Find the mean and the standard deviation of cross-training shoes Year Aerobic Basketball Cross- walking training A jar of peanut butter contains 457 grams with a standard deviation of 10.2 grams. Assuming a normal distribution, find the probability that a jar contains less than 454 grams Suppose a brand of light bulbs is normally distributed, with a mean life of 1100 hr and a standard deviation of 100 hr. Find the probability that a light bulb of that brand lasts between 950 hr and 1280 hr Assume that the total cholesterol levels for adults are normally distributed with mean cholesterol level of 51.4 mg/dl and standard deviation 14.3 mg/dl. Find the probability that an individual will have a cholesterol level greater than 60 mg/dl Key 1. $ $ % 4. 3 years 5. $31, % %, 4.303%, First Bank 8. $26,986

17 9. $23, years 11. $13, A) $251, B) $403, A) $ B) $ $ $182, $382.60, $3, A) $2, B) $301, C) $596, $16, $21, $10, $10, $109, $21, $ $11, $97, $21, $8, $12, $84, $21, $7, $14, $70, x + y 20, x 15, y Minimum is 18, no maximum 22. There is no minimum, maximum is pounds of cheese and 1.5 pounds of meat for a cost of $ small truck loads and 6 large truck loads 25. The maximum is z = 250 when x = 0, y = 50, s 1 = 30, s 2 = 0 and s 3 = The maximum is 6 when x = 3, y = 0, s 1 = 9, S 2 = 0 and s 3 = The maximum is 38 when x = 19, y = 0, z = 0, s 1 = 0, S 2 = Maximum is f = 216 at x = 72, y = 24 and z = Maximize f = 14x + 16y Subject to 2x + 3y x + 2y x + y x 0, y one speed, 0 three speed, and 2165 ten speeds for a maximum profit of $64, Basic Sets, 0 Regular Sets, 400 Deluxe Sets for a profit of $34000

18 32. z = 588, x = 49, y = f = 99, x = 33, y = 0, z = w = 360, x = 0, y = w = 20, x = 0, y = 5/4, z = {10,11,13} 37. A % 40. a) 7 b) 7 c) 32 d) 18 e) a) 511 b) 372 c) 2557 d) ¾ / ½ 45. a) 5/6 b) 5/6 c) 5/ / a) 0.37 b) 0.81 c) 0.79 d) a) b) c) d) : : / /

19 59. x P(x) $ $ $14, A $70.36, B $76.52, choose A a) 55,200 b) 70,304 c) ,760, , , , a) 165 b) ,036, a) 53,721,360 b) 74, / a) 1/1716 b) 1/1,235, a) 1008/4199 b) 3/8398 c) 8395/ a) b) c) a) b)

20 Interval frequency mpg lbs, mean: 17, median: 15, mode: range: 79, standard deviation: mean: , standard deviation: