a. Using the life expectancies, create a frequency distribution and a cumulative frequency distribution using 5 intervals.

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1 Name: JMJ Precalculus Honors Comprehensive Exam May 18, :40 AM 11:40 AM Mr. Casalinuovo Part 1: Statistics. Answer 3 of the following 4 problems from this part. Each problem is worth 7 points for a total of 21 points. 1) Below is a table that displays 22 countries, the number of televisions per person living in country in 2010, and the life expectancy of a person born in each country in 2010: Country People Life Exp country People Life Exp per TV per TV Angola Mexico Australia Morocco Cambodia Pakistan Canada Russia China 8 70 South Africa Egypt Sri Lanka France Uganda Haiti United Kingdom 3 76 Iraq United States Japan Vietnam Madagascar Yemen a. Using the life expectancies, create a frequency distribution and a cumulative frequency distribution using 5 intervals. Intervals Frequency Cumulative Frequency b. Draw a frequency histogram and a cumulative frequency histogram for the life expectancies. FREQUENCY HISTOGRAM CUMULATIVE FREQUENCY HISTOGRAM Xmin = XMax = XScl = YMin = YMax = YScl = (label each axis of the cumulative histogram) c. Find the percentile rank of the number of people per television

2 for Vietnam among the 22 countries. 2) Below is a table that gives the number of people injured in car accidents per 100,000 passengers traveling in cars for the years 1993 through 2009 in the United States: United States 1,863 1,685 1,744 1,766 1,700 1,571 1,557 1,499 1,367 1,324 1,297 1,221 1,164 1,869 3,805 1,682 1,631 a) Complete a five number summary for the data. Identify which pieces of information are involved and give the values. Round to the nearest tenth if necessary. Data Value b) Create a box and whisker plot for the data. c) Calculate the lower and upper fences for the data and dentify the outliers if any. Lower Fence = Upper Fence = Outlier(s) = d) Create a modified box plot for the data. e) Remove all outliers and recalculate the mean of the data. By how much does the mean change? (Round to the nearest tenth)

3 3) A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (seconds) to complete the escape ( Oxygen Consumption and Ventilation During Escape from an Offshore Platform, Ergonomics (1997) a) Construct a stem-and-leaf display of the data. Stem Leaf b) Calculate the mean, median, and standard deviation of the data. Round to the nearest tenth. Mean = Median = Standard Deviation = c) What percent of the data lies outside of 1 standard deviation from the mean? Round to the nearest percent.

4 4) Answer both a) and b) a) The Smith and Jones families each have six family members. They wanted to compare the differences in ages between the two families. The ages of the members of Smith family are 45, 43, 13, 11, 5, and 2, while the Jones family members are 45, 39, 17, 16, 4, and 1. Find the mean absolute deviation and range in ages for each family. Round to the nearest tenth. SMITHS JONES S Smith Family MAD = Jones Family MAD = Smith Family Range = Jones Family Range = Which family has the greatest difference in ages and why? b) The table below shows the test grades for three Algebra students. Test 1 counts for 10 % of the final average, test 2 counts for 20 %, test 3 counts for 30 %, and the final test will count for the rest. Test Bob Faye Cheryl Test Test Test Final Test Find each student s final average, rounding to the nearest tenth. Bob = Faye = Cheryl =

5 Part 2: Regression Analysis: You must answer all parts to this question. It is worth 10 points. The accompanying table shows wind speed and the corresponding wind chill factor when the air temperature is 10ºF. a) Use your graphing calculator to create a scatter plot of the data. Make a graph of the scatter plot in the accompanying viewing window. b) Use the regression feature of your graphing calculator to find a linear, quadratic, and a cubic model of the data. Graph each with the scatter plot, and round to three decimal places. LINEAR QUADRATIC CUBIC a = a = a = b = b = b = r = c = c = r = d = r = c) Which model best fit the data? (circle one) LINEAR QUADRATIC CUBIC d) Using the best fit equation, find the wind chill factor, to the nearest degree, when the wind speed is 50 miles per hour. d) e) Using the best fit equation, find the wind speed, to the nearest mi/h, when the wind chill factor is -18 F. e)

6 Part 3: Matrices. Answer all parts of this question. This question is worth 15 points. A = 3-8 2/3 B = C = 1/2-4 D = E = Evaluate (if possible): 1) 3C + 2D T 2) E -1 (In fraction form) 1) 2) 3) E X D 4) E x C T 3) 4) 5) Find the determinant of E. * This problem is continued on the next page.

7 6) Solve using Cramer s Rule.! 2x + 5y - 3z = 16 $ # & x - y - z = - 5 # " 3x + 2y + z = - 20 & % D = Dx = Dy = Dz = Solution = 7) For the problem, use the alphabet assignment in the table below. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Blank = 0 Decode the message : which was encoded with encoding matrix! # " $ &. % Message:

8 Part 4: Mortgage Problems. Answer all parts of this question. This question is worth 7 points. You and your spouse have narrowed down your search for a house to these two choices: Maple Avenue, Bluestown OR Spruce Lane, Redtown $ 379,999 $ 439,000 taxes = $ 12,600 taxes = $ 8,900 The bank that is giving you the mortgage requires a 25 % down payment and is offering a 30-year mortgage at 4.25 % interest. You wish to determine which house has the LOWER MONTHLY PAYMENT TO THE BANK. a) Calculate each house s down payment: Maple Avenue = Spruce Lane = b) Calculate each house s monthly payment not including taxes: Maple Avenue = Spruce Lane = c) Calculate each house s monthly payment including taxes: Maple Avenue = Spruce Lane = d) Which house has the cheaper monthly payment including taxes? e) How much in interest will you be paying over the 30-year mortgage by choosing the house you chose in part d)?

9 Part 5: Graphing Word Problems Answer 2 of the following 3 problems. Each problem is worth 5 points for a total of 10 points. 1) A projectile is fired vertically upward from a height of 600 feet above the ground, with an initial velocity of 803 ft/sec. Its height can be modeled by the equation: H = - 16x x where H is the height of the projectile and x is the length of time the projectile is in the air. (a) Graph the equation in an appropriate viewing window. Xmin = Xmax = Ymin = Ymax = (b) During what time interval will the projectile be more than 5000 feet above the ground? (Round to the nearest tenth of a second) (c) How long will the projectile be in flight? (Round to the nearest tenth of a second) (d) Find the maximum height of the projectile (to the nearest tenth of a foot).

10 2) The average speed, y (in knots) of the Trident motor yacht for different engine speeds x (in revolutions per minute (rpms)) can be approximated by the equation: y = x x x 9.53 a) Graph the equation in a viewing window displaying the average speed in knots from 0 to 50 rpms. Xmin = Xmax = Ymin = Ymax = b) Estimate the average speed of the Trident for an engine speed of 24 rpms. (Round to the nearest knot) c) What engine speed produces a boat speed of 14 knots? (Round to the nearest rpm) d) Calculate the increase in speed to the nearest knot when the engine speed increases from 20 to 30 rpms.

11 3. The annual US consumption of beef and poultry, y, in millions of pounds can be approximated by the following equations where x corresponds to the year. y = x 1,405,160 à beef y = x 632,699 à poultry a) Graph both equations in a viewing window displaying the consumption from 1970 to present: Xmin = Xmax = Ymin = Ymax = b) Use the equations to approximate the consumption of each in 1990 and in Round to the nearest million pounds. Beef Poultry c) Find the year in which the annual US consumption of beef and poultry was the same. Round to the nearest year. d) To the nearest million pounds, find the difference in consumption in beef and poultry in the year 2000.

12 Part 6: Graphing functions Answer 1 of the following 2 problems in this part. Answer all parts to the question you choose to answer. The problem is worth 11 points. 1) POLYNOMIAL EQUATION: Y = x 4-6x 3 + 3x 2 + 6x - 4 a) Graph the equation in a viewing window that allows for the viewing of the intercepts and the extrema points. Xmin = Xmax = Ymin = Ymax = b) Find the y-intercept of the equation to the nearest tenth if necessary. c) Find the zeros of the equation to the nearest tenth if necessary. d) Find the relative extrema (maximum / minimum point of the equation) rounding to the nearest tenth if necessary. Minimum(s): Maximum(s):

13 2) RATIONAL EQUATION: y = x 2 + 4x + 4 x 2 x - 6 a) Find the vertical asymptote(s). a) b) Find the horizontal or slant asymptote. b) c) Identify any holes. c) d) Find the x-intercept(s). d) e) Find the y-intercept. e) f) Sketch the graph in a viewing window that displays the asymptotes, holes, and the intercepts. Xmin = Xmax = Ymin = Ymax =

14 Part 7: Inequality Problems: Answer 2 of the following 3 problems. Each problem you choose to answer is worth 8 points for a total of 16 points. 1) A toy company makes toys at two different plants, Plant A and Plant B. Plant A needs to make a minimum of 1,000 toy dump trucks and fire engines in total. Plant B needs to make a minimum of 800 toy dump trucks and fire engines in total. Plant A can make 10 toy dump trucks and 5 toy fire engines per hour. Plant B can produce 5 toy dump trucks and 15 toy fire engines per hour. It costs $ 30 per hour to produce toy dump trucks and $ 35 per hour to produce toy fire engines. How many hours should be spent on each toy to minimize the cost and what is the minimum cost? Objective Formula: Constraints: Graph: X Min X Max Y Min Y Max VERTICES OF SOLUTION AREA Amount of Dump Trucks = Amount of Fire Engines = Minimum Costs =

15 2) A farmer has a 500-acre field in which she can plant wheat and corn. Each acre of wheat she plants will require 1 person-day of labor and has other expenses of $ 20. Each acre of corn will require 5 person-days of labor and has other expenses of $ 30. The farmer has $ 11,400 and 1,480 person-days of labor available, and she expects to make a profit of $ 90 per acre of wheat and $ 120 per acre of corn. How many acres of each should be planted to maximize her profit? Objective Formula: Constraints: Graph: X Min X Max Y Min Y Max VERTICES OF SOLUTION AREA Acres of Wheat = Acres of Corn = Maximum Profit =

16 3) The demand equation for a certain product is: p = x and the supply equation for the same product is: p = x where p is the price of the product and x is the number of units. Graph both the consumer and the producer surpluses on the same graph and calculate both. Xmin = Xmax = Ymin = Ymax = Point of Equilibrium = Consumer Surplus Inequalities Producer Surplus Inequalities Consumer Surplus: Producer Surplus:

17 PART 8: Exponentials You must answer all questions in this part. This part is worth 10 points. 1. The rodent population of a city is currently 30,000. It follows the Law of Uninhibited Growth and expects to double every 5 years. a) Find the exponential growth constant k to four decimals. Xmin = Xmax = Ymin = Ymax = a) k = b) Approximate the number of rodents in 8 years.. b) c) Approximate the number of rodents in 15 years. c) d) To the nearest tenth of a year, when will the rodent population reach 1,000,000. d) This problem continues on the next page.

18 2. $ 15,000 is invested at 5.6 % compounded quarterly. Find the balance after 6 years. 3. $ 35,000 is invested at 1.25 % compounded continuously. Find the balance after 5 years. 2) 3) 4. How long will it take an investment of $ 3,000 to triple when it is invested at 4⅜ % compounded monthly? (Round to the nearest tenth of a year) Xmin = Xmax = Ymin = Ymax = 4)