Conics Sequences And Series Probabillity

89 Systems of Nonlinear Equations Exercises 1 In the following exercises solve the system of equations by using graphing 2 In the following exercises solve the system of equations by using substitution 3 In the following exercises solve the system of equations by using elimination 4 In the following exercises solve the problem using a system of equations a The sum of the squares of two numbers is 25 The difference of the numbers is 1 Find the numbers b The difference of the squares of two numbers is 45 The difference between the square of the first number and twice the square of the second number is 9 Find the numbers c The perimeter of a rectangle is 58 meters and its area is 210 square meters Find the length and width of the rectangle 101 Terminology Student Outcomes Students determine the sample space for a chance experiment Given a description of a chance experiment and an event students identify the subset of outcomes from the sample space corresponding to the complement of an event Given a description of a chance experiment and two events students identify the subset of outcomes from the sample space corresponding to the union or intersection of two events Students calculate the probability of events defined in terms of unions intersections and complements for a simple chance experiment with equally likely outcomes Sumary Sample Space The sample space of a chance experiment is the collection of all possible outcomes for the experiment Event An event is a collection of outcomes of a chance experiment For a chance experiment in which outcomes of the sample space are equally likely the probability of an event is the number of outcomes in the event divided by the number of outcomes in the sample space Some events are described in terms of or and or not Exercises Consider a second scenario card that Alan created for his game Scenario Card 2 Tools Spinner 1 Spinner 2 a spinner with six equal sectors Place the number 1 in a sector the number 2 in a second sector the number 3 in a third sector the number 4 in a fourth sector the number 5 in a fifth sector and the number 6 in the last sector Directions chance experiment Spin Spinner 1 and spin Spinner 2 Record the number from Spinner 1 and record the number from Spinner 2 Five Events of Interest Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 Player Scoring Card for Scenario 2 Turn Outcome from Spinner 1 Outcome from Spinner 2 Points 1 2 3 4 5 1 Prepare Spinner 1 and Spinner 2 for the chance experiment described on this second scenario card Recall that Spinner 2 has six equal sectors 2 What is the sample space for the chance experiment described on this scenario card 3 Based on the sample space determine the outcomes and the probabilities for each of the events on this scenario card Complete the table below Event Outcomes Probability Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 4 Assign the numbers 15 to the events described on the scenario card Five Events of Interest Scenario 2 Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 5 Determine at least three final scores based on the numbers you assigned to the events Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 6 Alan also included a fair coin as one of the scenario tools Develop a scenario card Scenario Card 3 that uses the coin and one of the spinners Include a description of the chance experiment and descriptions of five events relevant to the chance experiment Scenario Card 3 Tools Fair coin head or tail Spinner 1 Directions chance experiment Five Events of Interest 7 Determine the sample space for your chance experiment Then complete the table below for the five events on your scenario card Assign the numbers 15 to the descriptions you created Event Outcomes Probability 8 Determine a final score for your game based on five turns Turn Points 1 2 3 4 5 91 Sequences Student Outcomes Students recognize when a table of values represents an arithmetic or geometric sequence Patterns are present in tables of values They choose and define the parameter values for a function that represents a sequence Summary A sequence is a list of numbers or objects in a special order An arithmetic sequence goes from one term to the next by adding or subtracting the same value A geometric sequence goes from one term to the next by multiplying or dividing by the same value Looking at the difference of differences can be a quick way to determine if a sequence can be represented as a quadratic expression Exercises Solve the following problems by finding the functionformula that represents the 𝑛th term of the sequence 1 After a knee injury a jogger is told he can jog 10 minutes every day and that he can increase his jogging time by 3 minutes every two weeks How long will it take for him to be able to jog one hour a day 2 A ball is dropped from a height of 15 feet The ball then bounces to 80 of its previous height with each subsequent bounce a Explain how this situation can be modeled with a sequence Week Daily Jog Time 1 10 2 10 3 13 4 13 5 16 6 16 b How high to the nearest tenth of a foot does the ball bounce on the fifth bounce 3 Consider the following sequence 8 17 32 53 80 113 a What pattern do you see and what does that pattern mean for the analytical representation of the function b What is the symbolic representation of the sequence 4 Arnold wants to be able to complete 100 militarystyle pull ups His trainer puts him on a workout regimen designed to improve his pullup strength The following chart shows how many pullups Arnold can complete after each month of training How many months will it take Arnold to achieve his goal if this pattern continues Month PullUp Count 1 2 2 5 3 10 4 17 5 26 6 37 103 Probability events Student Outcomes Students represent events by shading appropriate regions in a Venn diagram Given a chance experiment with equally likely outcomes students calculate counts and probabilities by addingsubtracting given counts or probabilities Students interpret probabilities in context Sumary In a probability experiment the events can be represented by circles in a Venn diagram Combinations of events using and or and not can be shown by shading the appropriate regions of the Venn diagram The number of possible outcomes can be shown in each region of the Venn diagram alternatively probabilities may be shown The number of outcomes in a given region or the probability associated with it can be calculated by adding or subtracting the known numbers of possible outcomes or probabilities Exercises 1 On a flight some of the passengers have frequentflier status and some do not Also some of the passengers have checked baggage and some do not Let the set of passengers who have frequentflier status be 𝐹 and the set of passengers who have checked baggage be 𝐶 On the Venn diagrams provided shade the regions representing the following instances 1 Passengers who have frequentflier status and have checked baggage 2 Passengers who have frequentflier status or have checked baggage 3 Passengers who do not have both frequentflier status and checked baggage 4 Passengers who have frequentflier status or do not have checked baggage 2 For the scenario introduced in Problem 1 suppose that of the 400 people on the flight 368 have checked baggage 228 have checked baggage but do not have frequentflier status and 8 have neither frequentflier status nor checked baggage 1 Using a Venn diagram calculate the following 1 The number of people on the flight who have frequentflier status and have checked baggage 2 The number of people on the flight who have frequentflier status 2 In the Venn diagram provided below write the probabilities of the events associated with the regions marked with a star 3 When an animal is selected at random from those at a zoo the probability that it is North American meaning that its natural habitat is in the North American continent is 065 the probability that it is both North American and a carnivore is 016 and the probability that it is neither American nor a carnivore is 017 1 Using a Venn diagram calculate the probability that a randomly selected animal is a carnivore 2 Complete the table below showing the probabilities of the events corresponding to the cells of the table North American Not North American Total Carnivore Not Carnivore Total 4 This question introduces the mathematical symbols for and or and not Considering all the people in the world let 𝐴 be the set of Americans citizens of the United States and let 𝐵 be the set of people who have brothers The set of people who are Americans and have brothers is represented by the shaded region in the Venn diagram below This set is written 𝐴 𝐵 read 𝐴 intersect 𝐵 and the probability that a randomly selected person is American and has a brother is written 𝑃𝐴 𝐵 The set of people who are Americans or have brothers is represented by the shaded region in the Venn diagram below This set is written 𝐴 𝐵 read 𝐴 union 𝐵 and the probability that a randomly selected person is American or has a brother is written 𝑃𝐴 𝐵 The set of people who are not Americans is represented by the shaded region in the Venn diagram below This set is written 𝐴𝐶 read 𝐴 complement and the probability that a randomly selected person is not American is written 𝑃𝐴𝐶 Now think about the cars available at a dealership Suppose a car is selected at random from the cars at this dealership Let the event that the car has manual transmission be denoted by 𝑀 and let the event that the car is a sedan be denoted by 𝑆 The Venn diagram below shows the probabilities associated with four of the regions of the diagram 1 What is the value of 𝑃𝑀 𝑆 2 Complete this sentence using and or or 𝑃𝑀 𝑆 is the probability that a randomly selected car has a manual transmission is a sedan 3 What is the value of 𝑃𝑀 𝑆 4 Complete this sentence using and or or 𝑃𝑀 𝑆 is the probability that a randomly selected car has a manual transmission is a sedan 5 What is the value of 𝑃𝑆𝐶 6 Explain the meaning of 𝑃𝑆𝐶 93 Binomial Theorem Student Outcomes Students observe patterns in the coefficients of the terms in binomial expansions They formalize their observations and explore the mathematical basis for them Students use the binomial theorem to solve problems in a geometric context Exercises 1 Consider the binomial 2𝑢 3𝑣6 a Find the term that contains 𝑣4 b Find the term that contains 𝑢3 c Find the third term 2 Consider the binomial 𝑢2 𝑣36 a Find the term that contains 𝑣6 b Find the term that contains 𝑢6 c Find the fifth term 3 Find the sum of all coefficients in the following binomial expansion a 2𝑢 𝑣10 b 2𝑢 𝑣10 c 2𝑢 3𝑣11 d 𝑢 3𝑣11 e 1 𝑖10 f 1 𝑖10 g 1 𝑖200 h 1 𝑖201 4 Expand the binomial 1 2𝑖 6 5 Show that 2 2𝑖 20 2 2𝑖 20 is an integer 6 We know 𝑢 𝑣2 𝑢2 2𝑢𝑣 𝑣2 𝑢2 𝑣2 2𝑢𝑣 Use this pattern to predict what the expanded form of each expression would be Then expand the expression and compare your results a 𝑢 𝑣 𝑤2 b 𝑎 𝑏 𝑐 𝑑2 7 Look at the powers of 101 up to the fourth power on a calculator Explain what you see Predict the value of 1015 and then find the answer on a calculator Are they the same 8 Can Pascals triangle be applied to 1 𝑢 1 𝑣 𝑛 given 𝑢 𝑣 0 9 The volume and surface area of a sphere are given by 𝑉 4 3 𝜋𝑟3 and 𝑆 4𝜋𝑟2 Suppose we increase the radius of a sphere by 0001 units from 𝑟 to 𝑟 0001 a Use the binomial theorem to write an expression for the increase in volume 𝑉𝑟 0001 𝑉𝑟 as the sum of three terms b Write an expression for the average rate of change of the volume as the radius increases from 𝑟 to 𝑟 0001 c Simplify the expression in part b to compute the average rate of change of the volume of a sphere as the radius increases from 𝑟 to 𝑟 0001 d What does the expression from part c resemble e Why does it make sense that the average rate of change should approximate the surface area Think about the geometric figure formed by 𝑉𝑟 0001 𝑉𝑟 What does this represent f How could we approximate the volume of the shell using surface area And the average rate of change for the volume g Find the difference between the average rate of change of the volume and 𝑆𝑟 when 𝑟 1 10 The area and circumference of a circle of radius 𝑟 are given by 𝐴𝑟 𝜋𝑟2 and 𝐶𝑟 2𝜋𝑟 Suppose we increase the radius of a sphere by 0001 units from 𝑟 to 𝑟 0001 a Use the binomial theorem to write an expression for the increase in area volume 𝐴𝑟 0001 𝐴𝑟 as a sum of three terms b Write an expression for the average rate of change of the area as the radius increases from 𝑟 to 𝑟 0001 c Simplify the expression in part b to compute the average rate of change of the area of a circle as the radius increases from 𝑟 to 𝑟 0001 d What does the expression from part c resemble e Why does it make sense that the average rate of change should approximate the area of a circle Think about the geometric figure formed by 𝐴𝑟 0001 𝐴𝑟 What does this represent f How could we approximate the area of the shell using circumference And the average rate of change for the area g Find the difference between the average rate of change of the area and 𝐶𝑟 when 𝑟 1