Mathematical Reasoning Questions and Answers

script 3 A researcher is interested in studying the difference in the proportion of Penn State undergraduates who attend graduate school upon graduation and the proportion of University of Oklahoma OU undergraduates who attend graduate school upon graduation Specifically the researcher believes the proportion of Penn State students who attend graduate school is higher than the proportion of OU students who attend graduate school To test her theory she randomly samples 250 Penn State seniors and finds 55 of those sampled will go directly to graduate school upon graduation In a random sample of 320 OU seniors 40 indicate they will go directly to graduate school upon graduation At the 01 level of significance is there enough information to conclude that the proportion of Penn State seniors who go directly to graduate school upon graduation is greater than the proportion of OU seniors who go directly to graduate school upon graduation List and clearly label all eight steps

5 The EPA has been worried about phosphorus concentration in Lake Erie They implemented an expensive environmental regulation in 2020 which was fully implemented in 2022 Phosphorus levels were sampled at eight randomly selected locations throughout the lake prior to the law s passage in 2020 and after implementation in 2022 The results in parts per million ppm are listed in the table below At the 01 level of significance test to see if the passage of the expensive new law has been successful in reducing lake phosphorus levels List and clearly label all eight steps Round to 4 FOUR decimal places Location 1 567AWN2 3 4 8 Before 4 30 2 66 5 67 7 07 7 16 6 32 5 89 4 69 After 4 15 2 47 5 97 6 57 7 31 6 51 5 64 4 40

MY NOTES Three different bacteria are cultured in one environment and feed on three nutrients Each individual of species I consumes 1 unit of each of the first and second nutrients and 2 units of the third nutrient Each individual of species II consumes 2 units of the first nutrient and 2 units of the third nutrient Each individual of species III consumes 2 units of the first nutrient 3 units of the second nutrient and 5 units of the third nutrient If the culture is given 5100 units of the first nutrient 7500 units of the second nutrient and 12 600 units of the third nutrient how many of each species can be supported such that all of the nutrients are consumed Let x species I y species II and z species III If there are infinitely many solutions express your answers in terms of z as in Example 3 x y z where 2400 z 2500

June 2014 Edith babysits for x hours a week after school at a job that pays 4 an hour She has accepted a job that pays 8 an hour as a library assistant working y hours a week She will work both jobs She is able to work no more than 15 hours a week due to school commitments Edith wants to earn at least 80 a week working a combination of both jobs Write a system of inequalities that can be used to represent the situation Graph these inequalities on the set of axes below y 20 10 10 20 per Determine and state one combination of hours that will allow Edith to earn at least 80 while working no more than 15 hours week

6 In basketball you can earn 3 points for a shot and 1 point for a free throw If Sam s probability of getting a 3 point shot is 4 10 and 8 10 for a free throw what is his expected value for the game

Ellipse with vertices at 3 7 and 3 9 and minor axis Length of 6

The toy machine in the shape of a rectangular prism at Six Flags contains balls The part that holds the balls is 32 in by 32 in and 48 inches tall One of the balls is 6 inches in diameter The machine is usually only halfway full How many balls can the machine hold if it is halfway full V lwh 4 V 77 3 You may use one or both formulas to solve the problem For the toolbar press ALT F10 PC or ALT FN F10 Mac M Arial BIVS Paragraph ST T99 20 BBY EX EBEB V 10pt BAEXE C8B13 EY T VAV O V I X0 Q32 EX X 4 10 poin O WORDS POWERED

48 m The arch of the bridge below is half an ellipse a semi ellipse As shown in the figure below the arch is 148 m long and has a height of 48 m at the center The point p is located 10 m from the center Find the height of the arch above point p Round your answer to the nearest hundredth Do not round any intermediate computations 10 m center 148 m 7 P 74 00m 12 X 13 M

7 Quadrilateral QRST is inscribed in the circle Complete the steps in the following logical argument that shows that the opposite angles of an inscribed quadrilateral are supplementary A 90 B 180 C 360 D 1 E 2 blank 1 blank 2 MTSR mRQT blank 3 blank 1 blank 2 blank 3 blank 4 mTSR mRQT mzQ 2 mTSR m s mRQT m2Q mes blank 4 R LA II B III C IV D V E T

18 Given AD and EF are diameters and BG and BF are tangents If mZFBD 25 which is false DOOO E MLGBD 25 G mLBCF 65 F mzDCF 115 H mLECG 25 OE F G H O B C E

Option 2 Find the measure of angle b a N e W D b d C

he table below gives the population of Texas since 1970 1970 1980 1990 2000 2010 Year Population in millions 11 2 14 2 17 0 20 9 25 1 A Find the average rate of change for each decade B During which decade was the average rate of change the largest C Use the average rate of change during the decade 1990 to 2000 to approximate the Texas population in 1994 D Use the average rate of change during the decade 2000 to 2010 to approximate the Texas population in 2008

The profit function on a product is given by P x 0 012 3x 172 where is the number of units sold and produced and P z is the total profit in thousands of dollars from selling and producing a units a How many units should be sold to maximize profits 057 33 units 0 0 02 units b Find the maximum 053 dollars O275 3 dollars O 150 units O 300 units profits that can be generated O278 dollars O 37 dollars

Rate of Change Guided Practice The table below gives the population of California since 1970 1970 1980 1990 2000 2010 Year Population in millions 20 0 23 7 29 8 33 9 37 3 A Find the average rate of change for each decade During which decade was the average rate of change the largest SLOPE rise change in y change in x run Given 2 points X1 Y1 X2 Y2 Formula Y2 1 X2 X1 B Use the average rate of change during the decade 1990 to 2000 to approximate the California population in 1993 C Use the average rate of change during the decade 2000 to 2010 to approximate the California population in 2009

Shortest Route Algorithm If you start at vertex A and use the Shortest Route Algorithm what would be the very first edge that would be selected in the very first step E B AD A D 11 C 13 12 3

Determine the equation of the parabola that opens down has focus 0 12 and a focal diameter of 20

Some people might not answer their phone after repeated calls O People with unlisted phone numbers or people with cell phones might be left out of the sample Some people might refuse to tell a stranger on the phone about their flossing habits Some people might feel pressured to say they floss even though they don t QUESTION 10 Which of the data distributions shown below has a higher variability Distribution A doans in the City floss daily Which of these is the best example of response bia Distribution B

You are organizing your high school s sports banquet The banquet hall rental is 250 In addition to this one time charge the meal will cost 7 per plate Let x the number of people who attend Part a Write an equation that represents the total cost C Part b Write an equation that represents the average cost A per person Part c Use the graph of the model shown below to estimate the number of people who must attend before the average cost drops to 15 per meal Part d Describe what happens to the average cost as the number of people who attend increases AA 100 90 80 70 60 50 40 30 20 10 X 5 10 15 20 25 30 35 40 45 50 For the toolbar press ALT F10 PC or ALT FN F10 Mac of Jont M H EA V T E

Construct a truth table for the Staten rv p q Fill in the missing values in the table Fill in the blanks below P T T T T F FL T L F F T T r T LL T F T F r P EV 2 D 4 P V p q

Construct a truth table for the given statement p q Fill in the table PT T In 1 q F p q

Let p q and r represent the following simple statements P The job pays well q I study r It is snowing outside Write the symbolic statement in words q r Ap Choose the correct answer below NINE OA If I study then it is snowing outside and the job pays well OB If the job pays well then I study or it is snowing outside OC If I study then it is snowing outside and the job pays well OD If the job pays well then it is snowing outside or I study

Let p and q represent the following statements p The chair is broken q The table is fixed Write the compound statement p q in words Choose the correct answer below OA The chair is broken if and only if the table is not fixed OB If the table is not fixed then the chair is broken OC If the chair is broken then the table is fixed OD If the chair is not broken then the table is fixed

Indicate whether the statement is a simple or a compound statement If it is a compound statement indicate whether it is a negation conjunction disjunction conditional or biconditional by using both the word and its appropriate symbol Statement Boise is a city in Idaho and is the capital of Idaho Is the above statement simple or compound 4 OA The statement is a compound statement because it has only one idea OB The statement is a compound statement because it combines two or more simple statements OC The statement is a simple statement because it has only one idea OD The statement is a simple statement because it combines two or more simple statements If the statement is compound is it a negation conjunction disjunction conditional or biconditional Select the correct choice below and if necessary fill in the answer box to complete your choice OA The statement is a biconditional It is represented by the symbol OB The statement is a negation It is represented by the symbol OC The statement is a disjunction It is represented by the symbol OD The statement is a conditional It is represented by the symbol O E The statement is a conjunction It is represented by the symbol OF The statement is simple

Decide whether the following statement is compound If it is a compound statement indicate whether it is a negation conjunction disjunction conditional or biconditional by using both the word and its appropriate symbol Jerry s wife loves swimming for fun Is the statement a compound statement O No Yes correct If the statement is compound is it a negation conjunction disjunction conditional or biconditional Choose the correct answer below OA negation OB disjunction V OC The statement is simple OD biconditional SPRE OE conjunction A OF conditional www

A study was done on people s body masses in grams and depicted in a histogram Estimate the mean mass Show your work 140 of Individuals 120 100 2 60 40 20 O 160 170 180 190 200 Body Mass g 210 220 230

a b 0 a b 2 QUESTION 8 4 6 8 10 What was the median amount of time spent studying Did the same number of students spend 0 4 hours studying as students studying 7 9 hours Explain Make comparative box plots both on one line for the following situation The entire senior class at Harrison High School was polled on how many hours a day spend of twitter You and your friend decide to poll your own classes to compare You and your friend s data are as follows From your class 3 1 2 1 0 6 4 0 From your friend s class 2 Minimum Q1 Median Q3 Maximum 2 12 2 6 3 Give the five number summary of each set of data Your Class 1 2 3 Your Friend s Class 1 How do these classes compare Discuss variation spread of data 2 1 2

The histogram describes the height of students in a math class Is the data univariate or bivariate Is the skewness left right or normal Hours Spent Playing Video Games on Weekends Number of students R I 5 9 99 1 84 98 15 19 99 Hours spent playing video games 20 24 99 Z

O Bar Chart O Stem and leaf plot QUESTION 5 sker Plot The box plot shows the number of students in different mathematics classes at a school Students in Different Mathematics Classes a b C 12 16 20 24 28 32 14 22 24 28 31 What is the median number of students in mathematics classes at the school How many students are in the smallest mathematics class How many students are in the largest math class Do at least half of the math classes have an enrollment of 25 or less Explain QUESTION 6 Which shows the quartiles and median most clearly O Box and Whisker plot O Histogram

The boxplot below shows the precipitation in San Diego CA in inches over the past 30 years In how many of those years was the rainfall between 10 3 and 19 4 inches Precipitation in San Diego California 0 1 1 6 5 years 10 years 7 5 years 15 years 13 5 10 H 20 3 7 8 1 10 3 12 8 19 4 1 10 QUESTION 2 The boxplot below shows the precipitation in Chicago IL in inches over the past 30 years Find the interquartile range of the data Precipitation in Chicago Illinois 0 20 I 1 30 26 6 30 40 I 40 50 32 0 36 2 40 1 50 I 49 4

An old medical textbook states that the mean sodium level for healthy adults is 140 mEq per liter of blood A medical researcher believes that because of modern dietary habits the mean sodium level for healthy adults now differs from that given in the textbook A random sample of 11 healthy adults is evaluated The mean sodium level for the sample is 149 mEq per liter of blood It is known that the population standard deviation of adult sodium levels is 12 mEq Assume that the population is normally distributed Can we conclude at the 0 05 level of significance that the population mean adult sodium level differs from that given in the textbook Perform a two tailed test Then complete the parts below Carry your intermediate computations to three or more decimal places and round your responses as specified below If necessary consult a list of formulas a State the null hypothesis Ho and the alternative hypothesis H Ho H 0 b Determine the type of test statistic to use Choose one c Find the value of the test statistic Round to three or more decimal places 0 d Find the p value Round to three or more decimal places e Can we conclude that the population mean adult sodium level differs from that given in the textbook H X 4 0 a X S 00 P 0 0 OSO Q G olo 0 0

For a fundraising project your math club is publishing a calendar The cost of the digital images and the permission to use them is 125 In addition to these one time charges the unit cost of printing each calendar is 3 25 Part a Write an equation for the total cost of the calendars Part b Write a model that gives the average cost A per calendar as a function of the number of calendars printed Part c Use the graph of the model shown below to estimate the number of calendars that must be sold before the average cost drops to 5 per calendar 25 22 5 20 17 5 151 12 5 10 7 5 055 in in 25 10 20 30 40 50 60 70 80 90 100 toolbar press ALT F10 PC or ALT FN F10 Mac T Y n t

1 point What is the negation of the following statement I exercise and I feel tired A I don t exercise or I feel tired OB I don t exercise or I feel envigorated OC I don t exercise and I feel envigorated O D I exercise and I feel tired OE I exercise and I feel envigorated OF I don t exercise and I feel tired O G I exercise or I feel envigorated OH I exercise or I feel tired What is the negation of the following statement I exercise or I feel tired OA I don t exercise and I feel tired OB I exercise and I feel tired C I don t exercise or I feel tired OD I don t exercise and I feel envigorated OE I exercise or I feel envigorated OF I exercise or I feel tired OG I exercise and I feel envigorated I fool onvigorated

Select the converse inverse and contrapositive for the following statement If Mexico is an island then Australia is an island The converse of the given statement is which of the following O If Mexico is not an island then Australia is not an island Mexico is not an island or Australia is an island O O If Australia is not an island then Mexico is not an island O If Australia is an island then Mexico is an island The inverse of the given statement is which of the following OIf Australia is an island then Mexico is an island If Mexico is not an island then Australia is not an island If Australia is not an island then Mexico is not an island O Mexico is not an island or Australia is an island The contrapositive of the given statement is which of the following OIf Mexico is not an island then Australia is not an island O Mexico is not an island or Australia is an island OIf Australia is not an island then Mexico is not an island O If Australia is an island then Mexico is an island

Use the fact that p q is equivalent to p q to write the statement in an equivalent form It is not true that if Carol went to the library then she read a book Which of the following is an equivalent form of the statement O A Carol went to the library and she read a book O B Carol did not go to the library and she read a book OC Carol did not go to the library and she did not read a book OD Carol went to the library and she did not read a book

1 Each block contains either 1 or 2 angles with different degree meas Type a 1 or 2 in the box next to the colored block to explain Which blocks have only 1 unique angle Which have 2

What is the correct adjacency matrix for the graph shown A B C D E F A 1 00 1 01 B002 1 00 CO 20 1 00 D 1 1 1 0 1 1 B000 1 0 1 F1 0 0 1 1 0 A B C D E F A 200 1 0 1 B002 1 00 CO 20 1 00 D1 1 1 01 1 EO 00 1 0 1 F1 00 10 A B C D E F A 200 1 0 BO 01 1 0 0 CO 1 0 1 0 0 D 1 1 0 1 E000 01 1 D F E A D B C

Suppose that the length of time between consecutive high tides is 12 hours and 25 minutes According to the National Oceanic and Atmospheric Administration c particular day in a city high tide occurred at 12 25 AM 0 42 hours and low tide occurred at 7 05 AM 7 08 hours Water heights are measured as the amounts al or below the mean lower low water The height of the water at high tide was 5 94 feet and the height of the water at low tide was 0 04 foot Complete parts a through c C a Approximately when will the next high tide occur The next high tide will occur at b Find a sinusoidal function of the form y A sin x B that models the data y sin x Simplify your answers Round to the nearest thousandth as needed c Use the function found in part b to predict the height of the water at 6 PM The height of the water at 6 PM is approximately ft Simplify your answer Type an integer or decimal rounded to the nearest thousandth as needed

Read the following excerpt from N Scott Momaday s House Made of Dawn Which of the following literary devices does the eagle function as in the excerp He had seen a strange thing an eagle overhead with its talons closed upon a snake It was an awful holy sight full of magic and meaning theme hyperbole symbol

Let S t be the number of daylight hours on the tth day of the year in Miami We are given the following data for S t Day January 25 March 30 May 26 July 8 f t We wish to fit a trigonometric function of the form 2 f t a b sin t 25 89 146 13 189 14 S t 11 12 2 365 365 to these data Find the best approximation of this form using least squares c cos

Question 3 25 points In the following graph each vertx has been labeled with its EST B START 0 C O ABCDEF O ABDF O ABEF O ACEF 5 D CI E What is the critical path if the minimum finish time is 20 hours F Question 4 25 points In the following Earliest Start Time ESP graph 2

Given ui j v 10i 4j and w 8j find proju v W

Scientist claims that pneumonia causes weight loss in mice The table shows the weights in gra mice before infection and two days after infection At a 0 10 is there enough evidence to sup e scientist s claim Assume the samples are random and dependent and the population is norm stributed House Weight before Weight after 1 2 3 4 5 6 21 1 22 8 20 6 19 9 20 3 20 4 19 6 22 0 19 8 19 5 19 2 19 4 OA Pneumonia causes weight loss in mice OB Weight gain causes pneumonia in mice OC Weight loss causes pneumonia in mice OD Pneumonia causes weight gain in mice Which calculator command would be most appropriate for testing the claim A T Test OB 2 Sample T Test OC Z Test Sempla

Computer Specialists is planning a group vacation to one of the following locations Alaska A Florida F San Antonio S or Hawaii H The employees rank the four possible sites according to the following preference table Number of Votes 5 7 5 6 8 A A H F F S H F S S S A F A H A H If the four voters who voted F S H A in that order change their votes to H S A F and if the plurality with elimination method is used to determine the winner is the monotonicity criterion satisfied O O 7 4 H F First A Second H S S Third F A H Fourth S F Yes No

Suppose that the plurality with elimination method is used on the following preference table If the 5 voters who voted A C B in that order change their votes to C A B determine if the monas criterion satisfied Number of Votes First Second Third 3 A B C Is the monotonicity criterion satisfied O Yes SBCA 6 5 ACB CCAB 10

The p value of an upper tail one tailed test is the area to the right of the calculated t value under appropriate t curve True False

0 30 A sample of 31 U S adults is chosen Use Excel as needed Part 1 of 5 a Is it appropriate to use the normal approximation to find the probability that more than 47 of the people in the sample have high blood pressure If so find the probability If not explain why not It is not appropriate to use the normal curve since np 9 3 Part 1 5 Part 2 of 5 10 b A new sample of 83 adults is drawn Find the probability that more than 41 of the people in this sample have high blood pressure Round the answer to at least four decimal places The probability that more than 41 of the people in this sample have high blood pressure is Espa ol 8 E

Darlena has started taking photos at amateur dog racing events later offering the photos for sale to the dog owners by email The prices she has charged per photo at each of her first three events and the corresponding number of photos sold and total revenue raised appear in the table below Price per Photo 30 27 18 Revenue Treating revenue as a function of the number of photos sold a graph of the three data points is also shown If she uses quadratic regression to fit a curve to the data what number of photos sold and what price per photo will maximize her revenue 1501 120 Number of Photos Sold 4 5 8 90 Revenue 120 135 144 Enable Zoom Pan

Show that the differential x y 0 0 is closed Show that it is not independent of path on any annulus centered at 0 Solution r P OP 2 y y 2y dy x y udr ady x y x y Pdx Qdy x y2 x y Because OP Jy JQ Or the differential is closed and we can use Green s Theorem Q x y x y x 21 x y y Y f f ydx zdy dx dy 1 1 2 1 1 1 1 dx dy 2 r 2 0 72 z r Thus the line integral is not independent of path

K A resort plans to apportion 49 new toboggans among the four resorts based on the number of rooms in each resort as shown below Use the table to answer parts a through c Resort A D 265 Total 637 Number of rooms 93 a Determine the standard divisor B 137 13 Type an integer or decimal rounded to the nearest hundredth as needed b Determine each resort s standard quota Resort Number of rooms Allocated toboggans Type whole numbers Resort A B 137 Total 637 Number of rooms 93 7 15 10 54 10 92 20 38 49 Standard quota Type integers or decimals rounded to the nearest hundredth as needed c Determine each resort s apportionment using Hamilton s method A 93 C 142 B 137 C 142 C 142 D 265 D 265 Total 637 49

A boat manufacturer has 147 new boats of a new model to be apportioned to four dealerships The manufacturer decides to apportion the boats based on the number of boats each dealership sold in the previous year The number of boats sold by each dealership is shown below Use the table to answer parts a through c Dealership A Annual Sales 4998 C D Total B 3596 2872 2058 13 524 a Determine the standard divisor 92 Type an integer or decimal rounded to the nearest hundredth as needed b Determine each dealership s standard quota Dealership Annual sales Allocated boats Type whole numbers Dealership A 4998 B 3596 Annual sales 22 37 Standard quota 54 33 39 09 Type integers or decimals rounded to the nearest hundredth as needed c Determine each dealership s apportionment using Hamilton s method A 4998 C 2872 31 22 B 3596 D 2058 C 2872 Total 13 524 D 2058 Total 13 524 B