Name Date. Key Math Concepts

Transcription

1 5-1 Classified Ads Sales tax = price of item sales tax rate Key Math Concepts A piecewise function gives a set of rules for each set of the function. The domain is defined by the inequalities that follow when. Guided Exercises 1. Enrique plans to sell his car and places a 6-line ad. His paper charges $42 for the first two lines and $6.75 per extra line to run the ad for one week. What will Enrique s ad cost to run for three weeks? Number of lines over 2: 6 2 = Cost of extra lines = number of extra lines cost of each extra line Cost of extra lines = = Total cost of ad = number of weeks(cost of the first two lines + cost of extra lines) Total cost of ad = ( + ) = The total cost of Enrique s car ad is. 2. The Fort Salonga News charges $29.50 for a classified ad that is four or fewer lines long. Each line above four lines costs an additional $5.25. Express the cost of an ad algebraically as a piecewise function. There are two rules: 1) the charges for ads 2) the charges for ads above For rule 1; c(x) = For rule 2; c(x) = + (x 4) when when The piecewise function is: when cx ( )= + ( ) when 64 Financial Algebra Guided Practice Workbook 5-1 or duplicated, or posted to a publicly accessible website, in whole or in part.

2 Exercises 3. Ms. Boyrer is writing a program to compute ad costs. She needs to enter an algebraic representation of the costs of a local paper s ad. The charge is $32.25 for up to three lines for a classified ad. Each additional line costs $6. Express the cost of an ad f(x) with x lines as a function of x algebraically. 4. Roxanne set up the following split function which represents the cost of an auto classified ad from her hometown newspaper when x 5 f( x) = ( x 5) when x > 5 If x is the number of lines in the ad, express the price c(x) of a classified ad from this paper in words. 5. Dr. Mandel purchased a used car for $11,325. Her state charges 8% tax for the car, $53 for license plates, and $40 for a state safety and emissions inspection. How much does she need to pay for these extra charges, not including the price of the car? 6. Graph the piecewise function: c(x) = { when x (x 6) when x > 6 What are the coordinates of the cusp? What is the slope of the graph where x > 6? What is the slope of the graph where x < 6? 7. Express the classified ad rate, $36 for the first four lines and $4.25 for each additional line, as a piecewise function. Use a Let statement to identify what x and c(x) represent. or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

3 5-2 Buy or Sell a Car Key Math Concepts Q1 is the median of the lower half of the data, Q2 is the median, Q3 is the median of the upper half of the data, and Q4 is the maximum value of the data set. IQR = Q3 Q1 To determine outliers, use Upper boundary = Q (IQR) and Lower boundary = Q3 1.5(IQR) Guided Exercises 1. Find the mean, median, mode, and range for the data set 34, 56, 44, 200. Mean = sum of the data number of items Mean = ( ) 4 = = 4 Median: write the numbers in ascending order:,,, The number of items is even so frind the average of the middle two numbers. Median: ( ) 2 = Range = greatest number least number = = The mean is, the median is, the range is, and there is no mode. 2. The data below gives the MPG ratings for cars owned by 15 Placid High School seniors. How many outliers are in this data set? 15.9, 17.8, 21.6, 25.2, 31.1, 29, 28.6, 32, 34, 14, 19.8, 19.5, 20.1, 27.7, 25.5 In ascending order:,,,,,,,,,,,,,, Q2 = the median = Q1 = the median of the lower half = Q3 = the median of the upper half = IQR = Q3 Q1 = = Upper boundary = Q (IQR) = + ( ) = Lower boundary = Q3 1.5(IQR) = ( ) = Any number above or below is an outlier. So, there are. 66 Financial Algebra Guided Practice Workbook 5-2 or duplicated, or posted to a publicly accessible website, in whole or in part.

4 Exercises 3. A local charity wants to purchase a classic 1956 Thunderbird for use as a prize in a fundraiser. They find the following eight prices in the paper. $48,000 $57,000 $31,000 $58,999 $61,200 $59,000 $97,500 $42,500 What is the best measure of central tendency to use to get a reasonable estimate for the cost of the car? Explain. 4. Find the value of x that will make the mean of the following data set equal to , 90, 88, 70, x Given is the list of prices for a set of used original hubcaps for a 1957 Chevrolet. They vary depending on the condition. Use the data for Exercises $120 $50 $320 $220 $310 $100 $260 $300 $155 $125 $600 $250 $200 $200 $ mean, to the nearest dollar 6. median 7. mode 8. four quartiles 9. range 10. interquartile range 11. boundary for the lower outliers; any lower outliers 12. boundary for the upper outliers; any upper outliers or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

5 5-3 Key Math Concepts Q1 is the median of the lower half of the data, Q2 is the median, Q3 is the median of the upper half of the data, and Q4 is the maximum value of the data set. IQR = Q3 Q1 Graph Frequency Distributions Guided Exercises 1. The back-to-back stem-and-leaf plot gives data about the number of girls (on the left) and boys (on the right) at each high school in Greene County who purchased their own cars with money they earned working. How many high schools are in Greene County? What is the mean number of boys who bought their own cars? Find the median of the distribution of girls. Count the pieces of data on one side. This is the number of schools. Number of schools = Mean = sum of the data number of items girls boys Mean (boys) = ( ) 10 = Median: The number of items is even. Find the average of the two middle numbers: Median = = 2 There are schools in Green County. A mean of boys bought cars and a median of girls bought cars. 2. The boxplot summarizes information about the numbers of hours worked in December for 220 seniors at Tomah High School. What is the interquartile range? What is the median? How many students worked 31 hours or less? IQR = Q3 Q1 = = Q2 is the median, which is the number The interquartile range is and the median is. of the data is before the median and is after the median. of the total number of students = = of the students work 31 hours or less. 68 Financial Algebra Guided Practice Workbook 5-3 or duplicated, or posted to a publicly accessible website, in whole or in part.

6 Exercises 3. Jerry is looking to purchase a set of used chrome wheels for his SUV. He found 23 ads for the wheels he wants online and in the classified ads of his local newspaper and arranged the prices in ascending order, which is given below. $350 $350 $350 $420 $450 $450 $500 $500 $500 $500 $600 $700 $725 $725 $725 $725 $725 $775 $775 $800 $825 $825 $850 Make a frequency table to display the data. Draw a box-and-whisker plot for the data. The side-by-side boxplots for distributions A and B were drawn on the same axes. Use the box plots to answer Exercises Which distribution has the greater range? A 5. Which distribution has the smallest interquartile range? B 6. What percent of the scores in distribution A is above distribution B s maximum score? 7. Which distribution has scores that are the most varied? 8. What percent of scores in distribution A are less than distribution B s first quartile? or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

7 5-4 Automobile Insurance Key Math Concepts 100/200/50 represents 100/200 BI insurance and 50,000 PD insurance. annual premium insurance payment = + surcharge number of payments per year Guided Exercises 1. Mr. Cousins has 100/300 bodily injury insurance. He was in an auto accident caused by his negligence. Five people were injured in the accident. They sued in court and were awarded money. One person was awarded $150,000, and each of the other two was awarded $95,000. How much will the insurance company pay for these lawsuits? 100/300 bodily insurance is $100,000 per person and $300,000 per accident. Of the people with injuries: 1 had $150,000 in injuries. $150,000 > $100,000, so insurance will pay $100, people had < $100,000, so insurance will pay $95,000 each. Add: + + = $290,000 < $300,000, so the insurance company will pay of these lawsuits. 2. The Chow family just bought a second car. The annual premium would have been a dollars to insure the car, but they are entitled to a 12% discount since another car is insured by the company. If they pay their premium semiannually, and have to pay a b dollars surcharge for this arrangement, express their semiannually payment algebraically. Discount = decimal annual premium = 0.12a Discounted premium = annual premium discount Discounted premium = = annual premium insurance payment = number of payments per year + surcharge a is the same as 1(a). Insurance payment = b Substitute the discounted premium. The Chow s discounted annual premium is and their payment is b. 70 Financial Algebra Guided Practice Workbook 5-4 or duplicated, or posted to a publicly accessible website, in whole or in part.

8 Exercises 3. Cai s annual premium is p dollars. If she pays her premium semiannually, there is a 1% surcharge on each payment. Write an expression for the amount of her semiannual payment. Jake has $25,000 worth of property damage insurance and $1,000-deductible collision insurance. He caused an accident that damaged a $2,000 sign, and he also did $2,400 worth of damage to another car. His car had $2,980 worth of damage done. Answer Exercises How much will the insurance company pay under Jake s property damage insurance? 5. How much will the insurance company pay under Jake s collision insurance? 6. How much of the damage must Jake pay for? 7. Allen Siegell has a personal injury protection policy that covers each person in, on, around, or under his car for medical expenses up to $50,000. He is involved in an accident and five people in his car are hurt. One person has $3,000 of medical expenses, three people each have $500 worth of medical expenses, and Allen himself has medical expenses totaling $62,000. How much money must the insurance company pay out for these five people? Mrs. Lennon has 100/275/50 liability insurance and $50,000 PIP insurance. During an ice storm, she hits a fence and bounces into a store front with 11 people inside. Some are hurt and sue her. A passenger in Mrs. Lennon s car is also hurt. 8. The store front will cost $24,000 to replace. There was $1,450 worth of damage to the fence. What insurance will cover this, and how much will the company pay? 9. The passenger in the car had medical bills totaling $20,000. What type of insurance covers this, and how much will the insurance company pay? or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

9 5-5 Guided Exercises Linear Automobile Depreciaton Key Math Concepts Straight line depreciation equation: y = mx + b, where y is the value of the car, x is the number of years, m is the slope of depreciation, and b is the original price of the car. Δx Slope ratio = Δy y-intercept: (0, maximum car value); x-intercept: (maximum lifespan, 0) Katie purchased a new car for $27,599. This make and model straight line depreciates for 13 years. 1. Identify the coordinates of the x- and y-intercepts for the depreciation equation. Determine the slope of the depreciation equation. y-intercept: (0, maximum car value) x-intercept: (maximum lifespan, 0) y-intercept: (, ) x-intercept: (, ) Δx Slope ratio = Δy Sloperatio = = = 0 13 Katie s new car depreciates at a rate of a year. 2. Write the straight line depreciation equation that models this situation. Draw the graph of the straight line depreciation equation. m = slope = b = y-intercept (original price of car) = y = mx + b y = x + Use the Y= function of the graphing calculator to create the graph. 72 Financial Algebra Guided Practice Workbook 5-5 or duplicated, or posted to a publicly accessible website, in whole or in part.

10 Exercises 3. The graph of a straight line depreciation equation is shown. Write the straight line appreciation equation. Use the graph to approximate when the car will be worth half its original value. $19,200 value $16,000 $12,800 $9,600 $6,400 $3, years 4. Caroline purchased a car four years ago at a price of $28,400. According to research on this make and model, similar cars have straight line depreciated to zero value after 8 years. How much will this car be worth after 51 months? 5. The straight line depreciation equation for a luxury car is y = 4,150x + 49,800. In approximately how many years will the car s value drop by 30%? 6. A new car sells for $29,250. It straight line depreciates in 13 years. What is the slope of the straight line depreciation equation? 7. A new car straight line depreciates according to the equation y = 1,875x + 20,625. What is the original price of the car? How many years will it take for this car to fully straight line depreciate? or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

11 5-6 Historical and Exponential Depreciation Key Math Concepts The exponential depreciation formula is y = A(1 r) x, where A is the car s standing value, x is the elapsed time in years, r is the percent of depreciation, and y is the value of the car after x years. Guided Exercises 1. Tanya s new car sold for $23,856. Her online research indicates that the car will depreciate exponentially at a rate of % per year. Write the exponential depreciation formula for Tanya s car. A is the starting value, or 23,856 r is the percent depreciation or, %, or. y = A(1 r) x Write 6 3 % as a decimal. 8 y = (1 ) x The exponential depreciation formula is y = (1 ) x. 2. Sharon purchased a used car for $24,600. The car depreciates exponentially by 8% per year. How much will the car be worth after 5 years? Round your answer to the nearest penny. A = r = or x = Write a depreciation equation. y = A(1 r) x y = (1 ) Substitute. y = Sharon s car will be worth after 5 years. 74 Financial Algebra Guided Practice Workbook 5-6 or duplicated, or posted to a publicly accessible website, in whole or in part.

12 Exercises 3. Brad purchased a five-year old car for $14,200. When the car was new, it sold for $24,000. Find the depreciation rate to the nearest hundredth of a percent. 4. A car exponentially depreciates at a rate of 8.5% per year. Mia purchased a 4-year-old car for $17,500. What was the original price of the car when it was new? Round your answer to the nearest thousand dollars. 5. The screen to the right is from a graphing calculator after running an exponential regression analysis of a set of automobile data. X represents years and y represents car value. Using the numbers on the screen, write the exponential regression equation. ExpReg y=a*b^x a= b= r 2= r= Nancy and Bob bought a used car for $22,800. When this car was new, it sold for $30,000. If the car depreciates exponentially at a rate of 9.2% per year, approximately how old is the car? Round your answer to the nearest hundredth of a year. 7. A new car sells for $31,400. It exponentially depreciates at a rate of 4.95% to $26,500. How long did it take for the car to depreciate to this amount? Round your answer to the nearest tenth of a year. or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

13 5-7 Driving Data Key Math Concepts D = R T, where D is distance, R is the rate, and T is time in hours. D = M G, M is miles per gallon, and G is the number of gallons used. Guided Exercises 1. Ruth is planning a 1,543-mile trip to a math teachers conference in San Diego. She plans to average 50 miles per hour on the trip. At that speed, for how many hours will she be driving? Express your answer to the nearest hour. D is the distance, or R is the miles per hour, or D = R T T = D R Rewrite to solve for T. T = Ruth will travel about hours. 2. Francois car gets about 12.5 kilometers per liter. He is planning a 1,600-kilometer trip. About how many liters of gas should Francois plan to buy? Round your answer to the nearest liter At an average price of $1.18 per liter, how much should Francois expect to spend for gas? D is the distance, or M is the miles per gallon, or D = M G G = D M Rewrite to solve for G. G = The total cost will be = Francois will buy about and spend about. 76 Financial Algebra Guided Practice Workbook 5-7 or duplicated, or posted to a publicly accessible website, in whole or in part.

14 Exercises 3. A car travels at an average rate of speed of 60 miles per hour for hours. How many miles does this car travel? 4. A car travels m miles in t hours. Express its average speed algebraically. 5. Abby s car gets approximately 24 miles per gallon. She is planning a 1,200-mile trip. About how many gallons of gas should she plan to buy? At an average price of $4.20 per gallon, how much should she expect to spend for gas? 6. Raquel is driving from New York City to Daytona, Florida, a distance of 1,034 miles. How much time is saved by doing the trip at an average speed of 60 mph as compared with 55 mph? Round to the nearest half-hour. 7. Ace Car Rental charges customers $0.18 per mile driven. You picked up a car and the odometer read x miles and brought it back with the odometer reading y miles. Write an algebraic expression for the total cost Ace would charge you for mileage use on a rented car. 8. Juanita has a hybrid car that averages 44 miles per gallon. Her car has a 12-gallon tank. How far can she travel on one full tank of gas? or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

15 5-8 Driving Safety Data Key Math Concepts Braking distance = s2, where s is the speed of the car in miles per hour. 20 Total stopping distance = reaction distance + braking distance Guided Exercises 1. How many feet does a car traveling at 57 mph cover in one hour? in one minute? in one second? 57 5,280 = 300,960 feet 300, 960 ft 1 hr hr 5,016 feet 60 min = 5,016 ft min 1 min =83. 6 feet 60 s One mile = 5,280 feet One hour = 60 minutes One minute = 60 seconds A car traveling 57 mph covers in one hour, in one minute, and in one second. 2. Jerry is driving 35 miles per hour as he approaches a park. A dog darts out into the street between two parked cars, and Jerry reacts in about three-quarters of a second. What is his approximate reaction distance? 35 mph 5,280 ft = ft per hour 60 min = ft per min 60s sec ft per second Jerry s car is traveling about per second. Write a proportion where x is the distance traveled where the reaction time is 0.75 s x (0.75) = x = x Jerry s reaction distance is about. 78 Financial Algebra Guided Practice Workbook 5-8 or duplicated, or posted to a publicly accessible website, in whole or in part.

16 Exercises 3. How many feet does a car traveling at a mph cover in one hour? in one minute? in one second? 4. How many kilometers does a car traveling at 50 kph cover in one hour? How many meters does a car traveling at 50 kph cover in one hour? in one minute? in one second? 5. Manuel is driving 60 miles per hour on a state road with a 65 mph speed limit. He sees a fallen tree up ahead and must come to a quick, complete stop. What is his approximate reaction distance? What is his approximate braking distance? About how many feet does the car travel from the time he switches pedals until the car has completely stopped? 6. Anita is driving on the highway at the legal speed limit of 63 mph. She sees a police road block about 300 feet ahead and must come to a complete stop. Her reaction time is approximately 4 3 of a second. Is she far enough away to safely bring the car to a complete stop? Explain your answer. or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook

17 5-9 Accident Investigation Data Key Math Concepts S= 30 D f n or 30Dfn, where S is the speed of the car, D is the skid distance, f is the drag factor, and n is the braking efficiency written as a decimal S 15 fr, where r is the radius of the yaw mark. 2 C M r = +, where r is the radius of the yaw mark, C is the length of the chord, and 8M 2 M is the length of the middle ordinate. Guided Exercises 1. Mark was traveling at 40 mph on a road with a drag factor of His brakes were working at 80% efficiency. To the nearest tenth of a foot, what would you expect the average length of the skid marks to be if he applied his brakes in order to come to an immediate stop? S = 30Dfn D = 50 feet; f = 1.1; n = 0.85 S 30( )( )( ) Substitute. S mph The minimum speed of the car is mph. 2. Determine the radius of the yaw mark made when brakes are immediately applied to avoid a collision based upon a yaw mark chord measuring 59.5 feet and a middle ordinate measuring 6 feet. Round your answer to the nearest tenth. 2 C M r = + 8M 2 2 ( ) ( r = + 8( ) 2 r ) Substitute. The radius of the yaw mark is about. 80 Financial Algebra Guided Practice Workbook 5-9 or duplicated, or posted to a publicly accessible website, in whole or in part.

18 Exercises 3. A car is traveling at 52 mph before it enters into a skid. The drag factor of the road surface is 0.86, and the braking efficiency is 88%. How long might the average skid mark be to the nearest tenth of a foot? 4. Arami s car left three skid marks on the road after she slammed her foot on the brake pedal to make an emergency stop. The police measured them to be 45 feet, 40 feet, and 44 feet. What skid distance will be used when calculating the skid speed formula? 5. Andrea was traveling down a road at 45 mph when she was forced to immediately apply her brakes in order to come to a complete stop. Her car left two skid marks that averaged 60 feet in length with a difference of 6 feet between them. Her brakes were operating at 90% efficiency at the time of the incident. What was the possible drag factor of this road surface? What were the lengths of each skid mark? 6. The measure of the middle ordinate of a yaw mark is 7 feet. The radius of the arc is determined to be 64 feet. What was the length of the chord used in this situation? Round your answer to the nearest tenth of a foot. 7. Russell was driving on a gravel road that had a 20 mph speed limit posted. A car ahead of him pulled out from a parking lot causing Russell to immediately apply the brakes. His tires left two skid marks of lengths 60 feet and 62 feet. The road had a drag factor of His brakes were operating at 95% efficiency. The police gave Russell a ticket for speeding. Russell insisted that he was driving under the limit. Who is correct (the police or Russell)? Explain. or duplicated, or posted to a publicly accessible website, in whole or in part. Financial Algebra Guided Practice Workbook