Fundamentals of College Mathematics
Fundamentals of College Mathematics MATH 120
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This 11 page Class Notes was uploaded by Hoyt Beer on Sunday October 25, 2015. The Class Notes belongs to MATH 120 at University of Nevada - Las Vegas taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/228627/math-120-university-of-nevada-las-vegas in Mathematics (M) at University of Nevada - Las Vegas.
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Date Created: 10/25/15
Chapter 11 Section 1 Page 1 Section 111 The Fundamental Counting Principle Homework pg 585 problems 122 De nition The Fundamental Counting Principle is used to determine the total possible choices when combining groups of items If you can choose one item from a group of M choices and another from a group of N choices then the total number of twoitem choices is MN Example Checkpoint I A restaurant offers 10 appetizers and 15 main courses In how many ways can you order a twocourse meal Solution 10 15 150 choices Definition If you have more than two groups the Fundamental Counting principle still applies you just multiply the number of ways in which each event can occur these events must be exclusive Example Checkpoint 3 A pizza can be ordered with two choices of size medium or large three choices of crust thin thick or regular and ve choices of toppings beef sausage pepperoni bacon or mushroom How many different onetopping pizzas can be ordered Solution 235 30 Example Checkpoint 5 You are taking a multiplechoice test that has six questions Each of the questions has three answer choices with one correct answer per question If you select one of these three choices for each question and leave nothing blank in how many ways can you answer the questions Solution 333333 729 Chapter 10 Section 5 Page 1 Section 105 Volume Homework pg 549 l29 De nition Volume refers to the amount of space in cubic units occupied by a solid object The only measurements of volume you will be required to memorize are Volume of Rectangular solid area of bottom height length width height Volume of a Right Circular Cylinder area of bottom height 727 r2 h The other formulas you should be able to use if needed but do not have to be memorized Volume ofa pyramid 13 area of base height Volume ofa Cone 13 727 r2 h Volume ofa Sphere 43 727 r3 De nition The area of the outer surface of a three dimensional object is call surface area Again these formulas can be reasoned instead of memorized Surface area of Box top area bottom area side areas 21w 21h 2wh Surface area of Circular cylinder top area bottom area side area 2727 r2 2727 rh there is a typo in the book on this one it doesn t take into account the bottom area Chapter 12 Section 1 Page 1 Section 121 Sampling Frequency Distributions and Graphs Homework pg 605 problems 119 De nition Numerical information organized or listed is called data De nition Statistics is a way of collecting organizing analyzing and interpreting data as well as drawing conclusions based on the data De nition A population is the set containing all that is to be described and analyzed A sample is a subset of the population Special care must be taken when choosing the sample as it cannot be biased in any way Example If you want to nd out if individuals are against drinking alcohol you wouldn t ask only those citizens in a bar Example Checkpoint I A city government wants to conduct a survey among the city s homeless to discover their opinions about required residence in city shelters from midnight until 6am a Describe the population b A commissioner suggests obtaining a sample by surveying all the homeless people at the city s largest shelter Is this a good idea a The population is the city s homeless b No because those in a shelter are biased for them Solution De nition A random sample is a sample obtained in such a way that every element in the population has an equal chance of being selected for the sample Example Checkpoint 2 Why is Checkpoint 1b a bad way to sample and how can one accomplish this Solution It is a bad sample because those that are not in a particular shelter on a particular night have no chance of being selected Getting a random sample of homeless would be difficult to implement at best But if you could round them all up assign them a number and draw the numbers like a lottery Then they would all have an equal chance De nition A piece of data is called a data item and the value it takes is called a data value De nition Frequency refers to the number of times a particular data value occurs A frequency distribution organizes the data in terms of the frequency it lists all items with their corresponding frequency Chapter 12 Section 1 Page 2 0 Example Checkpoint 3 Construct a frequency distribution for the grades in precalc below FABBCCBCAACCDCB DCCBC Solution De nition Data can be arranged into groups called classes and then needs to be organized in a grouped frequency distribution Example Checkpoint 4 For the data below use the classes 4049 5059 6069 7079 8089 and 9099 and construct a grouped frequency distribution for the following exam scores 3 58 68 75 94 79 96 79 87 83 89 52 99 97 89 58 95 77 75 81 75 73 73 62 69 76 77 71 50 57 41 98 77 71 69 90 75 indicates a tick Solution T list Definition A histogram is a bar graph with data values or classes and their corresponding frequencies A frequency polygon is made from a histogram by connecting the middle of the bar with a line we will not spend class time doing these examples as they are not commonly used compared to the histogram 0 Example Construct a frequency histogram for Checkpoint 4 Solution 16 Frequenq 4049 5059 6069 7079 8089 9099 Test Scores Chapter 12 Section 3 Page 1 Section 123 Measures of Dispersion Homework pg 691 problems 132 De nition Measures of dispersion are used to describe the spread of data items in a data set De nition The range is the difference between the highest and lowest data values Range largest data value 7 smallest data value Example Checkpoint I Find the range for the data 24711 Solution The largest data value is 11 the smallest is 2 So the data range is 11 7 2 9 Definition The standard deviation is a measure of dispersion that depends on all the data items It is calculated by nding the overall difference between each data item and the mean To nd the standard deviation with n data items Calculate the mean of the data Subtract the mean from each data item Square each value above Add all the values from above Divide these values by n 7 1 Take the square root of the value above omeww Example Checkpoint 3 Find the standard deviation for the data 24711 Solution 2 4 7 1 1 6 1 The mean is we get 5 We have 4 items in the set so we divide by 3 9 1533 6 Taking the square root we nd the standard deviation is 392 Chapter 12 Section 3 Page 2 0 Example Checkpoint 4 Find the standard deviation for the data sets 737577798183 and 4044929498100 Solution 73 75 77 79 81 83 6 For the rst data set the mean is 78 25911925 The standard deviation is computed by f 374 4044929498100 78 For the second data set the mean is 6 98 7 400 14441156 196 256 400 484 6 J by V 2806 The standard deviation is r 5 This describes that the spread of data is more with the second set which is why it has a larger standard deviation Chapter 10 Section 1 Page 1 Section 101 7 Points Lines Planes and Angles Homework pg 513 12 536 De nition A point is represented as a small dot and usually written as a capitol letter pointA It has no dimension but merely specifies a place in space De nition A line connects two distinct points with the shortest possible distance in other words it is straight It goes on forever in both directions It is expressed with a lower case letter line I or using the letters of each point next to each other with a line above line g or EA De nition A plane is a at surface with no boundaries and it has no thickness It is two dimensional meaning you can move two different directions on that surface De nition We can take portions of lines A line going in one direction is a ray The ray has an initial point where it begins and a terminal point to specify where it heads It is expressed with the letters of the points ray or EA note the initial point has no arrow above it A line segment is just the portion of the line between two points known as endpoints It is expressed without arrowheads line segment AB or BA Note that lines and line segments can be written with either point first but for a ray you have to be careful that the initial point has no arrowhead above it De nition An angle is formed by two rays that meet at their initial points which is known as the vertex of the angle It has an initial side and a terminal side which can be difficult to determine out of context Most angles are written in standard position on the twodim ensional xy axis which is when the initial side is lined up with the positive xaxis and the vertex at the ordered pair 00 Angles are named several ways You can name the angle with three points the vertex B and one point on each ray AampC as iABC or 4 ABC or ABC Or you canjust use the vertex as 1B or 4 B or B Sometimes there is a greek letter inside the angle between initial and terminal sides and you can use that letter 15 or 4 B or lt1 3 You measure angles by finding the amount of rotation from the initial side to the terminal side Angles can be measured in degrees there are 360 in a circle or radians there are 2739 radians in one circle 7139 is approximately 314 Fractional components of degrees are minutes 60 minutes 1 degree or seconds 60 seconds 1 minute This is the same as time measurements why Chapter 10 Section 1 Page 2 There are special angles Right angle 90 degrees Straight angle 180 degrees Acute angle between 0 and 90 degrees Obtuse angle between 90 and 180 degrees There are special relationships between angles Complementary they sum to 90 degrees Supplementary they sum to 180 degrees Example Find the angle measures in the diagram using the de nition of supplementary angles Notice any new special relationships in the diagram Angles that are across a vertex from eachother are called vertical angles and their measures are the same When a parallel line is cut with another line all the angles that correspond A1 131 have the same measure These are called corresponding angles Chapter 10 Section 6 Page 1 Section 106 Rith Triangle Trig Homework pg 557 140 De nition Trigonometry is the study of the relationship between right triangles angles and sides We must rst be able to identify the parts of a triangle no matter what position it is in the labels remain the same opposite angle adjacent to angle The three fundamental trig lnctions are sin 9 opposite hypotenuse cos 0 adjacent hypotenuse tan 9 sin 9 cos 0 opposite adjacent Given any two sides of a right triangle you can nd the third side and nd the trig values for any angle Example Checkpoint I Find sine cosine and tan of 9 in the gure Solution The missing side l32 42 5 3 sin0 35 cos9 45 tan0 3A 4 Given any side of a right triangle and any angle other than 90 you can nd all other angles and all other sides Example Checkpoint 2 Find all pieces of the triangle given Solution 618079074050 c tan40 2 a 150tan4012586 40 deg 2 2 c 12586 150 l9581 150 Many application problems use angle of elevation which is the angle made up from the horizontal line of sight or angle of depression which is the angle made down from the horizontal line of sight Chapter 8 Section 4 Page 1 Section 84 Home Ownership Homework pg 460 14 0 De nition A mortgage is a long term loan used to buy a home 0 De nition The down payment is the amount of money paid at closing towards the purchase price of a house 0 There are xed rate mortgages refered to as xed meaning the interest rate is xed for the entire loan period and variable rate mortgages refered to as ARMs which vary depending on the market from year to year or even month to month 0 Typically one gets a loan from a mortgage broker This broker gives you the money to purchase the home minus the down payment at a given interest rate and charges you a certain fee known as a point or points One point is one percent of the mortgage amount 0 We do not need Table 84 We can nd these values using the formula we were given in our previous class To calculate the monthly payment principal plus interest for a home costingA dollars at i per month for a total of n months the monthly payment R is given by Ai 11 li Example The price of a home is 195000 You put 10 down and nance the rest with a 30 year xed mortgage at 75 APR What is your monthly payment and how much do you pay over the life of the loan Solution Loan amount A 195000 7 01195000 175500 i 007512 000625 and n 3012 360 Ai l75500000625 n 360 1 L 1 li 100625 Over the life ofthe loan you pay 122712 360 times 44176320 Your interest alone 44176320 7 175500 2662632 R 122712 0 Example Checkpoint I The price of a home is 240000 The bank requires a 10 down payment and three points at the time of closing The cost of the home is nanced with a 30 year xed mortgage at 65 APM Find the required down payment amount of mortgage amount due at closing to the lender the monthly payment and the total interest Solution Down payment 10 of 240000 24000 Amount ofmortgage 240000 7 24000 216000 Amount due at closing 3 points of 216000 3 of 216000 6480 216000000542 Monthly payment R 136527 360 1 100542 Total loan paid 360 136527 49149610 Total interest paid 49149610 7 216000 27549610 Chapter 8 Section 4 Page 2 There are many things to consider when qualifying for a mortgage 1 Your income 2 Your debts do you owe on a car do you have outstanding credit card bills 3 Your credit rating are you late paying bills have you claimed bankruptcy have you defaulted on a loan before All these things will be considered when applying for a loan Depending on the interest rate most mortgage company s will allow you to stretch to 3 times your debt to income ratio that is take your income and subtract your debts then multiply by 3 Example If I earn 50000 per year and owe 1500 per year on my automobile and another 500 per year on credit cards how much can I qualify for Solution Income 7 Debts per year 50000 7 1500 7 500 48000 I could qualify for as much as 144000 ho me So if you are considering buying a home the best thing you can do is eliminate all your debts and earn great credit pay off your credit cards pay off your car and pay all your bills on time But wait there s more Your mortgage payment consists of more than principal and interest You also have to pay 1 Taxes This can be estimated at 1 of your property s value per year but realtors should be able to tell you this with each property you look at 2 Insurance This can be estimated at 033 per year but once you have an address you can call any insurance company and they can give you a yearly quote 3 Flood Insurance The federal government makes maps that determine if you are in a ood zone Ifyou are you have to pay ood insurance This is not typical in vegas but you could get it anyway 4 Mortgage Insurance If you put less than 20 down on a home the mortgage company considers you risky and charges you mortgage insurance also known as PMI This rate will vary depending on how much principal you have invested but is usually about 05 per year Example What would be an estimate of my total monthly payment on the example above assume no ood insurance is needed Solution Amount of mortgtge 240000 7 24000 216000 PampI 2160000005462 136527 1 100542 Taxes per year 240000 00112 200 Insurance 240000 0003312 66 PMI 240000 0005212 100 Total Monthly Payment 136527 200 66 100 173127
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