2GeometryFormulas you need to know!See reference table
3Geometry Finding Perimeters In the diagram, ABCD is an isosceles trapezoid. Its bases are AB and CD. BA is extended to E, and DE and EB are perpendicular. Side BC is a diameter of semicircle O, AB = 4, AE = 3, DE = 4, and DC = 10. Find the perimeter of the figure to the nearest tenth.Perimeter = distance around the figureFind the distance around the semicirclea2 + b2 = c2= c225 = c25 = cIf AD = 5 then BC = 5Perimeter =Isosceles TrapezoidAD = BCPerimeter = …= 28.9 units
4Geometry Finding Areas Find the area of the composite figure pictured below. Represent your answer in terms of pi.Area of Triangle + Area of SemicircleA = ½ bhA = ½ (12)(14)A = 84Remember: An answer left in terms of pi is accurate and exact. Rounding leads to an approximate result. Never round unless otherwise directed.
5GeometryShaded AreaMr. Petri has a rectangular plot of land with a length of 20 feet and a width of 10 feet. He wants to design a flower garden in the shape of a circle with two semicircles at each end of the center circle, as shown in the accompanying diagram. He will fill in the shaded area with wood chips. If one bag of wood chips covers 5 square feet, how many bags must he buy?Area of Rectangle – Area 2 CirclesA = lwA = (10)(20)A = 200Diameter = 10Radius = 5Area of Shaded Region200 –…square feetBags of Wood Chips NeededMr. Petri will need 9 bags of wood chips to cover the shaded area.
6Geometry Surface Area and Volume V = lwh (not on the reference table) Find the volume of the cylinder to the nearest hundredth.V = lwh (not on the reference table)SA = 2lh + 2hw + 2lwFind the surface area to the nearest hundredth of the cylinder if it represents a can which has no lid or bottom.
7Relative ErrorAny measurement made with a measuring device is approximate. The error in measurement is a mathematical way to show the uncertainty in the measurement. It is the difference between the result of the measurement and the true value of what you were measuring. The relative error expresses the "relative size of the error" of the measurement in relation to the measurement itself.Always subtract smaller number from bigger number to create a positive difference.
8Relative Error = .04 The relative error is 0.2037 A student mistakenly measures the length of a radius to be 24 inches. The actual radius is 25 inches.Find the relative error.Find the percent of error.= .04Percent Error = .04 x 100 = 4%The groundskeeper is replacing the turf on a football field. His measurements of the field are 130 yards by 60 yards. The actual measurements are 120 yards by 54 yards. What is the relative error, to the nearest ten thousandth, in calculating the area of the football field?Actual AreaA = lwA = (120)(54)A = 6480Measured AreaA = lwA = (130)(60)A = 7800The relative error is
9Probability The Counting Principle If there are a ways for one activity to occur, and b ways for a second activity to occur, then there are a • b ways for both to occur. Multiply the number of ways each activity can occur.Examples: 1. Activities: roll a die and flip a coin There are 6 ways to roll a die and two ways to flip a coin. There are 6 • 2 = 12 ways to roll a die and flip a coin.2. Activities: a coin is tossed five times There are 2 ways to flip a coin when each coin is flipped. There are 2 • 2 • 2 • 2 •2 = 32 arrangements of heads and tails.
10Probability The Counting Principle A movie theater sells 4 sizes of popcorn (small, medium, large and extra large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased?ways of ordering popcornYour state issues license plates consisting of letters and numbers. There are 26letters and the letters may be repeated. There are 10 digits and the digits may benot be repeated. How many possible license plates can be issued with two lettersfollowed by three numbers?Letters: 26 (with repeats) Digits: 0 – 9 (10 total without repeats)license plates
11Probability Permutations 4 P4 = 4! A permutation is an arrangement of objects in a specific order. The order of the arrangement is important!! Consider, four students walking toward their school entrance. How many different ways could they arrange themselves in this side-by-side pattern?1,2,3,4 2,1,3,4 3,2,1,4 4,2,3,1 1,2,4,3 2,1,4,3 3,2,4,1 4,2,1,3 1,3,2,4 2,3,1,4 3,1,2,4 4,3,2,1 1,3,4,2 2,3,4,1 3,1,4,2 4,3,1,2 1,4,2,3 2,4,1,3 3,4,2,1 4,1,2,3 1,4,3,2 2,4,3,1 3,4,1,2 4,1,3,2The number of different arrangements is 24 or 4! = 4 • 3 • 2 • 1. There are 24 different arrangements, or permutations, of the four students walking side-by-side.The notation for a permutation: n Pr n is the total number of objects r is the number of objects chosenConsider the example above: There are 4 friends and all 4 friends are being arranged.4 P4 = 4!
12Probability Permutations 7 P7 = 7! 7 • 6 • 5 • 4 • 3 • 2 • 1 Find the number of ways to arrange 7 books on a shelf.7 P7 = 7!7 • 6 • 5 • 4 • 3 • 2 • 1= 5040 waysNot all permutations are factorials!Find the number of ways to arrange 5 books on a shelf chosen from a set 7 books.7 P5= 7 • 6 • 5 • 4 • 3= 2520 waysCalculator Corner:To compute factorials (!)…Example: 7!Enter number (7)Press MathScroll to the right to PRBPress #4 (!)EnterTo compute permutations (n Pr)…Example: 7 P5Enter the 1st number (7)Press MathScroll to the right to PRBPress #2 (n Pr)Enter second number (5)Enter
13Probability Theoretical & Experimental Probability Theoretical Probability of an event is the number of ways that the event can occur, divided by the total number of outcomes. Ex: What is the probability of landing on an even number if a die is rolled?Sample Space:Even #’s: 2, 4, 6Empirical Probability of an event is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials). It is based specifically on direct observations or experiences. Ex: Mary rolled a die 25 times and landed on an even number 9 times. What is the empirical probability that Mary will land on an even number on her next roll?
14Sum of the rolls of two dice ProbabilityTheoretical & Experimental ProbabilityKaren and Jason roll two dice 50 times and record their results in the accompanying chart. 1) What is their empirical (experimental) probability of rolling a 7? 2) What is the theoretical probability of rolling a 7?Empirical ProbabilityTheoretical ProbabilitySum of the rolls of two dice3, 5, 5, 4, 6, 7, 7, 5, 9, 10, 12, 9, 6, 5, 7, 8, 7, 4, 11, 6, 8, 8, 10, 6, 7, 4, 4, 5, 7, 9, 9, 7, 8, 11, 6, 5, 4, 7, 7, 4, 3, 6, 7, 7, 7, 8, 6, 7, 8, 9
15Probability Sample Spaces A sample space is a set of all possible outcomes for an activity or experiment.Ex: Marnie wants to choose an outfit consisting of a blouse (green or red), a pair of pants (jeans or khakis) and a pair of shoes (sandals or sneakers). Create a sample space to show all the different possible outfits she can make.Green Blouse, Jeans, SandalsGreen Blouse, Jeans, SneakersGreen Blouse, Khakis, SandalsGreen Blouse, Khakis, SneakersRed Blouse, Jeans, SandalsRed Blouse, Jeans, SneakersRed Blouse, Khakis, SandalsRed Blouse, Khakis, Sneakers8 possible combinations of outfitsHow many outfits include a pair of jeans?What is the probability that Marnie will choose an outfit with a red blouse or sneakers?4 outfits GB, Jeans, Sa GB, Jeans, Sn RB, Jeans, Sa RB, Jeans, SnRB, K, Sn GB, J, Sn RB, K, SaRB, J, Sn GB,K, Sn RB, J, Sa
16Probability Sample Spaces HHH HHT HTH HTT THH THT TTH TTT Sample spaces can also be represented using tree diagrams.Ex: Using a tree diagram, create the sample space for tossing a coin 3 times.How many outcomes include two heads and a tail?What is the probability of landing on at least two heads out of the three tosses?HHHHHTHTHHTTHHTHTHTHT3 outcomesTHHTHTTTHTTTHHHHHTHTHTHT4/8
17Probability Compound Probability If A and B are independent events, then P(A and B) = P(A) • P(B).“With Replacement”If A and B are dependent events, and A occurs first,then P(A and B) = P(A) • P(B, once A has occurred)“Without Replacement”Example: A drawer contains 3 red paperclips, 4 green paperclips, 5 blue paperclips, 1 white paperclip and 2 yellow paperclips. One paperclip is taken from the drawer and then replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue?If the first paperclip is not replaced, what is the probability that first paperclip is red and the second is blue?If the first paperclip is not replaced, what is the probability that both paperclips are red?Without Replacement: Denominator decreases!
18Probability Conditional Probability The conditional probability of an event B, in relation to event A, is the probability that event B will occur given the knowledge that an event A has already occurred.Example: You toss two pennies. The first penny shows HEADS and the other penny rolls under the table and you cannot see it. What is the probability that they are both HEADS? Sample Space-Tossing two Coins:HH TTHT TH outcomesBased on the information given, the sample space only includes HH and HT.
19Probability Conditional Probability Grade Snowboarding Skiing Middle school students were surveyed about what their favorite sport is. The results are shown in the following table. If a student is selected at random, what is the probability that the student prefers snowboarding given that he/she is in sixth grade grade?GradeSnowboardingSkiingIce SkatingTOTAL6th6841461557th8456702108th597447180211171163545Conditional Probability is the same as Conditional Relative FrequencyCondition: The student is in 6th gradeWhat is the probability that the student prefers snowboarding?
20Now it’s your turn to review on your own Now it’s your turn to review on your own! Use the information presented today to help you practice questions from the Regents Exams in the Green Book. See halgebra.org for the answer keys. Integrated Algebra Regents Review #3 Tomorrow (Tuesday), June 17th BE THERE!