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# Topics in Finite Mathematics MATH 210

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This 8 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 210 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/205286/math-210-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.

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Date Created: 09/17/15

Chapter 567 Problems Math 210 Spring 2006 Problem 1 Find the solution set for the fol lowing systems of simultaneous equations A 4x 7 5y 3 z 7 3y 5 z y 7 2x 7 y 5 z 7 3y 6 7 2m 6y 71 2 2x 7 3y 8 4x 7 6y 14 4x 7 y i 1 3x 7 2y 3 z 7y 8 7 5x 7 35y 7 40 z y 2 5x 7 3y 7 2 z 7 y 4 7 6m 6y 20 Suppose that Anne Becky and Carl have a total of 1500 Becky has 15 less than twice what Anne and Carl have to gether Anne has 3 less than half what Becky and Carl have together How much money do they each have Problem 2 Using the matrix de nitions be low preform the listed matrix operations If it is not possible explain why 123 13 A 456B11 71273 17 1 0 0 4 o g D 1 1 E 2 5 i lwl f S 3 a AB h F 1 b AC 1 ADE c AD j ABE72O d AE k FG e AF 1 GF f A71 m FE g 0 1 Problem 3 Solve the following problems in volving modeling by nding the technology matrix and applying a b Suppose that it takes unit of steel and 3 units of coal to produce one unit of steel Also it takes i unit of steel to produce one unit of coal Find the re quired amount of production to satisfy a demand of i 10 units of steel and 20 units of coal ii 100 units of steel and 200 units of coal Suppose that 10 units of steel and 15 units of coal are used to produce 20 units of steel Also 20 units of steel are used to produce 100 units of coal Find the required amount of production to satisfy a demand of i 10 units of steel and 20 units of coal7 ii 100 units of steel and 200 units of coal c Messy It takes 1 unit of steel and 1 unit of wood to make 4 units of coal It takes 15 units of coal7 4 units of steel7 and 8 units of wood to make 20 units of steel It takes 1 unit of coal and 1 unit of wood to make 2 units of wood Find the required amount of production to satisfy a demand of i 100 units of steel7 100 units of wood and 100 units of coal7 ii 100 units of wood and 200 units of coaL iii 300 units of steel Problem 4 Find the region speci ed by the following inequatlities Using the given func tion7 maximize and minimize the given func tion a z 2 0 y 2 0 7 7 i4y 2 8 Mfg 8x 3z7y S 9 z 2 0 y 2 0 7 7 y 21 Miz y 2z3y 6 z 2 0 y 2 0 7 7 yi 10 Mi2z y ziy S 15 z 2 0 y 2 0 7 7M S 8 M7z5y 5zi2y S 50 z 2 0 y 2 0 7 3y 2 6 M72zy 3z7y 2 24 z 2 0 y 2 0 f 2zy S 6 M4z5y y S 2 y 7 x S 1 Problem 5 Solve the following linear pro gramming problems a Your company makes footballs and base ball gloves One needs 2 pounds of leather and pound of stitching to make one football To make a baseball glove7 you need 4 pounds of leather and i pound of stitching You have 1000 pounds of leather and 100 pounds of stitching lf footballs go for 10 dollars apiece and baseball gloves for 15 dollars apiece7 how should you allocate your re sources to maximize your sales You employ for a Florida based company that hired graduates from the Univer sity of Florida UF and Florida State University FSU You can hire up to 100 people total Since UF is better school than FSU by a considerable mar gin7 UF graduates make more money and can make more product then their FSU counterparts Suppose that a UF grad can make 40 units of some product in a day and gets paid 200 a day An FSU grad can make 30 units in a day and get paid 160 a day If the company must keep labor costs below 18000 per day7 what is largest number of units the com pany can produce Chapter 489 Review Math 210 Spring 2006 Problem 1 Suppose you are out to the bar for the night If you drink a beer7 then there is a 09 chance you will order another beer and a 01 chance you will order a glass of water If you drink a glass of water7 there is a 01 chance of you ordering a beer7 a 07 chance of ordering another water7 and a 02 chance of going home a Set up a transition matrix for this sce nario with the states being ordering a beer7 ordering a water7 and going home b ls this Markov chain absorbing regular c If you started off drinking a beer at the bar7 how many beers would you expect to drink d If you started off with a water7 how many drinks would you have before you leave the bar Problem 2 Suppose you are on a beer pong team If you win your last game7 there is a 06 chance you will win the next game and a 04 chance of losing If you lose your last game7 then there is a 02 chance of winning your next game7 a 06 chance of losing your next game7 and a 02 chance of passing out from too much alcohol a Set up a transition matrix with three states7 winning the last game7 losing the last game7 and passing out b ls this Markov chain absorbing regular c Suppose you have an even chance of win ning or losing your rst game How many games do you expect to play Problem 3 Suppose you are playing an old school 8 bit Nintendo game You have a fa vorite game that you played so many times as a kid you can play it blindfolded Also7 you have a couple days before your nal to kill so you gure playing the game 100 times is a smart idea Suppose there is a 05 chance of you beating the game without dying and a 09 chance of you beating the game Let X be the number of wins overall and let Y be the number of wins without dying a What are the possible values for X and Y What are the expected values for X and Y What are the standard devia tions for X and Y Can the probabilities of X be approx imated with a standard distribution Can the probabilities of Y be approxi mated with a standard distribution Ex plain lf you can use standard distribution7 nd Pr45 S X S 60 If you can use standard distribution7 nd P7 Y 2 92 A D V Problem 4 At Papa John7s Pizza7 they use a technique they dub as Cheeselocking This is the idea that when making a pizza7 you should have the cheese layer over lap the crust and cover all of the sauce That way7 when the pizza cooks7 the cheese will be baked on to the crust and the toppings will be less likely to slide off the top of the pizza Suppose that an average Papa John7s store applies Cheese lock around 60 of the time On an average Friday night7 say that a store makes about 300 pizzas Let X be the number of pizzas with correct Cheese lock a What are the possible values of X What is the expected value of X What is the standard deviation of X Can we approximate with standard dis tribution Explain lf so7 what is Pr170 S X 185 Find PrlX S 175 In order to NOT be able to use a stan dard distribution in a situation with 300 pizzas made7 how good or how bad must the stores Cheese lock percent age be Problem 5 Suppose you are moving to the area from far away You may choose to rent an apartment7 buy a house7 or buy a condo If you are living in a house7 there is a 95 chance you will stay in a house and a 005 chance you will buy a condo If you are living in a condo7 there is a 07 chance you will stay in the condo and a 03 chance of living in an apartment If you are living in an apartment7 there is a 06 chance you will stay in the apartment7 a 03 chance of buying a condo7 and a 01 chance of buying a house a Find the transition matrix for this Markov chain b ls this regular Absorbing Explain c What are the stable probabilities of this Markov chain d If you have a 06 chance of renting7 a 02 chance of buying a house7 and a 02 chance of buying a condo7 what is the probability that after 4 years7 you will be living in a house A D V If you started by moving into an apart merit7 what is the expected number of years before you move into a house Problem 6 Monthly cell phone usage can be modeled by standard distribution The aver age number of minutes used per month with a cell phone is 600 minutes and the standard deviation for the number of minutes used is 50 a What is the probability of making ex actly 600 minutes of calls b What is the probability of making be tween 500 and 650 minutes of calls c What is the probability of making more than 700 minutes of calls d What is the probability of making less than 450 minutes of calls e If 40 of people use between 600 7a and 600 a minutes7 what is a f The top 20 heaviest cell phone users use at least how many minutes Problem 7 Suppose we are playing a dart game You start out with 2 dollars You throw a dart at the board and if you hit a single7 you lose a dollar If you hit a dou ble7 you net nothing If you hit triple7 you get one dollar If you hit the bulls eye7 you win the difference of your total and 4 dollars If you dont hit the board7 you lose all your money On a random throw7 you have a 05 chance of hitting a single7 a 02 chance of hit ting a double7 a 02 chance of hitting a triple7 a 005 chance of hitting a bulls eye7 and a 005 chance of missing the board a Find a transition matrix modeling this game where the states are the possible amounts of money you can have at any point in the game b ls this chain regular absorbing c What is the expected number of darts you would throw d What is the probability of winning all 4 dollars Problem 8 Suppose you buy a truck for 13000 dollars and nance at 39 com pounded monthly How much should monthly payments be to pay off the truck in a 3 years b 5 years Problem 9 You want to save 500000 and plan on doing this by putting away a cer tain amount every month in an account with 4 annual interest How much would install ments be if you wanted the total amount in a 10 years with interest compounded quar terly b 20 years with interest compounded monthly Problem 10 a If you invest money at 10 compounded daily by what proportion does your money increase over 8 years If you triple your money invested in 8 years what is the annual percentage yield If you invest at 7 compounded monthly and quadruple your money how many months did you invest d If you invest money at 10 compounded quarterly for 5 years and end up with 8000 dollars How much money did you initially need to invest Problem 11 Suppose you are going to pay for maintenence to your car at 100 per month If you plan to do this for 3 years and interest is 8 compounded monthly how much should you set aside Problem 12 Suppose you would like to drive a new BMW With the features you want you could buy one new for 35000 which you can get nanced at 39 for 10 years with monthly payments The other op tion is to lease the car for 350 per month The going interest rate for savings is 450 a How much would your payments be on the car if you buy it A C7 V What is the present value of the money spent leasing the car for 3 years A O V How much equity do you have in the car after 3 years After 3 years the expected value of the car will be 25000 Suppose after 3 years you sell the car and pay off the loan Would you have been better off leasing the car Chapter 567 Problems Math 210 Spring 2006 Problem 1 Find the solution set for the fol lowing systems of simultaneous equations a 95 3 5 ears b 2 f g g 43 C 72 1 71 9571 2 ltdgtizw e 3 23 1 77 f 75 i 3 40 41 g 5 f 33 3 e3 h 76 4 6g 23 Suppose that Anne7 Becky7 and Carl have a total of 1500 Becky has 15 less than twice what Anne and Carl have to gether Anne has 3 less than half what Becky and Carl have together How much money do they each have 357 Problem 2 Using the matrix de nitions be low7 preform the listed matrix operations If it is not possible7 explain why 123 13 A 456311 71273 17 7 16 77 14 29 710 c AD d AE Not Poss e AF Not Poss Daub 30MB f A 1 H Cth 3le m p g 0 1 Not Poss 11 23 77 l wiHMHooiH k FG 114 723 71 3 77 l GF Not Poss m FE 13 Problem 3 Solve the following problems in volving modeling by nding the technology matrix and applying a Suppose that it takes unit of steel and 3 units of coal to produce one unit of steel Also it takes i unit of steel to produce one unit of coal Find the re quired amount of production to satisfy a demand of 11 A 4 1 01 i 10 units of steel and 20 units of coal 120 200 ii 100 units of steel and 200 units of 1200 coal 2000 Suppose that 10 units of steel and 15 units of coal are used to produce 20 units of steel Also 20 units of steel are used to produce 100 units of coal Find the required amount of production to satisfy a demand of l l A 1 a a 1 Z 0 i 10 units of steel and 20 units of coal 40 50 ii 100 units of steel and 200 units of coal 40 39 500 Messy It takes 1 unit of steel and 1 unit of wood to make 4 units of coal It takes 15 units of coal 4 units of steel and 8 units of wood to make 20 units of steel It takes 1 unit of coal and 1 unit of wood to make 2 units of wood Find the required amount of production to satisfy a demand of MlQMquot O Wwih biw miH 3le i 100 units of steel 100 units of wood 400 960 ii 100 units of wood and 200 units of and 100 units of coal coal 240 776 1104 iii 300 units of steel 720 1128 Problem 4 Find the region speci ed by the following inequatlities Using the given func tion maximize and minimize the given func tion z 2 0 y 2 0 i 7 a i4y 2 78 Mfg 8x 37y S 9 Max 02 Min 43 m y b xy 2x3y Max 30 Min 02 0 a M7y 6 1 W W W 1 1 W W o HHOO 1 lt i2lt i2 z 2 0 y 2 0 fgy S 8 Miz5y 5x72y lt 50 Max 610 Min 0 0 z 2 0 y 2 0 i 3y 2 6 M72zy 3z7y 2 24 No Max Min13 z 2 0 y 2 0 f 2y S 6 M4x5y y S 2 117 S 1 Max 2 2 Min 0 0 Problem 5 Solve the following linear pro gramming problems a Your company makes footballs and base A C7 ball gloves One needs 2 pounds of leather and pound of stitching to make one football To make a base ball glove you need 4 pounds of leather and i pound of stitching You have 1000 pounds of leather and 100 pounds of stitching lf footballs go for 10 dol lars apiece and baseball gloves for 15 dol lars apiece how should you allocate your resources to maximize your sales 100 footballs 200 baseball gloves You employ for a Florida based company that hired graduates from the Univer sity of Florida UF and Florida State University FSU You can hire up to 100 people total Since UF is better school than FSU by a considerable mar gin UF graduates make more money and can make more product then their FSU counterparts Suppose that a UF grad can make 40 units of some product in a day and gets paid 200 a day An FSU grad can make 30 units in a day and get paid 160 a day If the company must keep labor costs below 18000 per day what is largest number of units the com pany can produce 50 of each

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