Basic Math Questions and Answers

1 Given that sin 3 14 cos 0 7818 use equivalent trigonometric expressions to evaluate the following

2 Use an appropriate compound angle formula to express as a single trig function and then determine an exact value T sin cos TE TL cos sin T

expectation when you throw the dart 9 a The probability that the dart lands on 5 is 16 Type a fraction Simplify your answer The probability that the dart lands on 10 is Type a fraction Simplify your answer The probability that the dart lands on 70 is Type a fraction Simplify your answer The probability that the dart lands on 100 is 100 1 16 W

Solve x In x y y 2 In x using integrating factors

8 a Find the missing values in the following data X 1 2 3 4 y 11 22 40 b Find the polynomial f x interpolating the following data 0 1 4 5 X f x www 5 140 2 2 6 7M

6 The determinant of a 2x2 matrix a b C Question Find the determinant of is ad bc 6 39 1 4 For more information on this topic look in your textbook at section 8 3 You might also want to look at the Matrix Operations presentation

O 30 O 12 O 15 27 auct The dot product of two vectors u a i b j and v a i b j is defined as u v a a2 b b For more information on this topic look in your textbook at section 6 7 You might also want to look at the Dot Product presentation Question How much is 3i 2j 7i 3j NOTE Two vectors are orthogonal perpendicular if and only if their dot product is zero

Assume that you are 28 years old today and that you are planning on retirement at age 65 Your current salary is 54 000 and you expect your salary to increase at a rate of 4 per year as long as you work To save for your retirement you plan on making annual contributions to a retirement account Your first contribution will be made on your 29th birthday and will be 8 of this year s salary Likewise you expect to deposit 8 of your salary each year until you reach age 65 Assume that the rate of interest is 9 The present value PV at age 28 of your retirement savings is closest to 71 196 60 133

Cost of Goods Sold Budget Delaware Chemical Company uses oil to produce two types of plastic products P1 and P2 Delaware budgeted 35 000 barrels of oil for purcha in June for 90 per barrel Direct labor budgeted in the chemical process was 240 000 for June Factory overhead was budgeted at 400 000 during June The inventories on June 1 were estimated to be Oil P1 15 200 8 300 P2 8 600 Work in process 12 900 The desired Inventories on June 30 were 16 100 9 400 7 900 13 500 Oil P1 P2 Work in process Use the preceding Information to prepare a cost of goods sold budget for June For those boxes in which you must enter subtracted or negative numbers use a minus sign Delaware Chemical Company Cost of Goods Sold Budget For the Month Ending June 30 Finished goods inventory June 1 Work in process inventory June 1 Direct materials Direct materials inventory June 1 Direct materials purchases Cost of direct materials available for use Direct materials inventory June 30 Cost of direct materials placed in production Direct labor Factory overhead Total manufacturing costs Total work in process during the period Work in process inventory June 30 Cost of goods manufactured h

Direct Labor Cost Budget Ace Racket Company manufactures two types of tennis rackets the Junior and Pro Striker models The production budget for July for the two rackets is as follows Junior 1 500 units Pro Striker 6 200 units Production budget Both rackets are produced in two departments Forming and Assembly The direct labor hours required for each racket are estimated as follows Forming Department Assembly Department 0 24 hour per unit 0 30 hour per unit Junior 0 16 bour per unit Pro Striker 0 20 hour per unit The direct labor rate for each department is as follows Forming Department Assembly Department 18 00 per hour 14 00 per hour Prepare the direct labor cost budget for July Feedback Ace Racket Company Direct Labor Cost Budget For the Month Ending July 31 Forming Department Hours required for production Junior Pro Striker Total Hourly rate Total direct labor cost 00000 Assembly Department 00000

A firm issues eightyear bonds with a coupon rate of 5 8 paid quarterly The credit spread for this firm s eightyear debt is 1 5 New eightyear Treasury notes are being issued at par with a coupon rate of 2 1 What should the price of the firm s outstanding eightyear bonds be per 100 of face value 115 23

Luther Corporation Consolidated Income Statement Year ended December 31 in millions 2022 610 1 500 2 109 9 40 5 Total sales Cost of sales Gross profit Selling general and administrative expenses Research and development Depreciation and amortization Operating income Other income Earnings before interest and taxes interest income pretax income Taxes Net income Price per share Sharing outstanding millions Stock options outstanding millions 24 6 3 6 41 2 41 2 25 1 16 1 5 5 10 6 16 10 2 0 3 Stockholders Equity Total Liabilities and Stockholders Equity mnt above Luther s return on equity ROE for the year ending December 31 2021 is closest to 126 6 533 1 2021 554 8 360 1 194 7 40 3 23 8 3 3 127 3 127 3 15 112 3 39 305 72 995 15 8 0 2 63 6 386 7

1 Complete each of the following difference columns Then indicate what type of graph the table would produce Justify your answer K 16 X 2 1 0 1 2 Type Reason y 1 3 9 27 81 1ST Diff X 2 1 0 1 2 Type Reason y 0 1 4 9 16 1ST Diff X 2 1 0 1 2 Type Reason y 8 3 2 7 12 1ST Diff

Use letters to represent each non negated simple statement and rewrite the given compound statement in symbolic form If my car is in working order then I can go to the store and I cannot go to the store Let p represent the simple sentence My car is in working order and q represent I can go to the store The compound statement written in symbolic form is

Let p and q represent the following simple statements p It is snowing outside q The job pays well Write the symbolic statement p q in words Choose the correct sentence below OA If it is snowing outside then the job does not pay well OB It is snowing outside if and only if the job does not pay well OC If it is snowing outside then the job pays well OD If the job does not pay well then it is snowing outside

Let p and q represent the following simple statements p I complete my homework q I go to the gym Write the following compound statement in symbolic form I complete my homework or I do not go to the gym The symbolic form is

Let p and q represent the following simple statements p This is an alligator q This is a reptile Write the following compound statement in symbolic form This is not an alligator if it s not a reptile The compound statement written in symbolic form is

Determine whether or not the sentence is a statement Quiet down Is the given sentence a statement Yes because it is a command O No because it is a command Yes because it could be labeled as true or false O It cannot be determined with further information

Construct a truth table for the given statement qv p Complete the truth table P T T q T F P qv p q v p

Let p and q represent the following simple statements p I m old q You re young Write the following compound statement in symbolic form You re not young but I m old The symbolic form is

Let p and q represent the following statements p Sue s dress is blue q Salvadore Dali did not paint the Sistine Chapel Express the following statement symbolically Salvadore Dali painted the Sistine Chapel The statement written symbolically is

Form the negation of the statement Monday does not come after Sunday Choose the correct answer below O A Sunday does come after Monday OB Sometimes Monday does not come after Sunday OC Monday comes after Sunday OD It is not true that Monday comes after Sunday

Number of students histogram below to answer the questions Annual Car Insurance Premiums for Students 650 701 Second class c How many students have an annual car insurance premium in the class with a class mark of 7523 students who have an annual car insurance premium in the class with a class mark of S 752 803 854 905 956 1007 Annual amount dollars There are d What is the class mark of the modal class The class mark of the modal class is e Construct a frequency distribution from this histogram Price 625 675 676 726 727 777 778 828 829 879 880 930 931 981 982 1032 Number of Students

Six people were asked to determine the amount of money they were carrying to the nearest dollar The results are shown below 32 62 12 26 3 93 a Determine the range and standard deviation of the amounts The range of the amounts is Simplify your answer m The standard deviation of the amounts is Round the final answer to the nearest cent as needed Round all intermediate values to the nearest cent as needed b Add 10 to each of the six amounts Determine the range and standard deviation of the new amounts The range of the new amounts is Simplify your answer The standard deviation of the new amounts is Round the final answer to the nearest cent as needed Round all intermediate values to the nearest cent as needed

Find the range and standard deviation of the set of data 4 8 9 11 12 16 The range is Simplify your answer GELD The standard deviation is Round the final answer to the nearest hundredth as needed Round all intermediate values to the nearest hundredth as

The accompanying table shows the ticket sales of the 15 highest grossing movies of all time adjusted for inflation in millions of dollars a Construct a frequency distribution with a first class of 877 996 b Construct a histogram c Construct a frequency polygon A Number of Movies 155442 Top Grossing Movies 8185 Sales n millions 1806 5 Top Grossing Movies c Choose the correct graph below OA h 2165 1806 3 C Sales in million B Number of Movies EXERCISE Top Grossing Movies 8165 OB Sales in millions Top Grossing Movies 1896 5 816 5 18065 Salmons OCCESS KILD bc Number of Movies gorff 816 5 O C Ticket sales in millions of dollars 1765 1640 1302 1277 1240 1165 1148 1118 1036 990 958 953 949 921 879 Top Grossing Movies Sales n millions Top Grossing Movies 18065 816 5 SU 18065 Sales in millions C OD Number of Move Top Crossing Movies 8165 Top Crossing Movies 165 18065 Sees in mors C

If the initial population of pandas was 100 and grew to 125 after the first year which of the following functions best models the population of pandas P with respect to the number of years t if the population growth of pandas is considered to be linear A P 25t 100 B P 100 1 25 C P 100 1 2 D P 201 5t 100 A m C

adjusted for inflation in millions of dollars a Construct a frequency distribution with a first class of 877 996 b Construct a histogram c Construct a frequency polygon a Construct a frequency distribution with a first class of 877 996 Sales million Number of Movies 877 996 DODP KODUN MOON MGAN HUN S HE 10 b Choose the correct graph below OA Top Grossing Movies 1896 sales of the 15 highest grossing movies of all time Q BUBUDOC Top Grossing Movies 8165 18065 2 GELED OC Ticket sales in millions of dollars 1640 1302 1277 1240 1036 990 921 879 1148 1118 949 8165 1765 1165 958 953 Top Grossing Movies Q Q OD Top Grossing Movies 8165 1096 C

If the initial population of rats was 20 and grew to 25 after the second year which of the following functions best models the population of rats P with respect to the number of years t if the population growth of rats is considered to be exponential A P 5t 20 B P 20 1 25 C P 20 5 D P 5t 20 A B

A project requires an initial investment of 60 000 and has cash inflows of 10 000 each year for 3 years What is the profitability index if the required return is 9 Would you accept this project

2 Determine the approximate degree measure to the nearest tenth K a 1 24 b 2 82

1 Determine the approximate radian measure to the nearest hundredth K a 33 b 347

A project has cash flows of 5 000 4 800 and 3 300 in years 1 2 and 3 If the cost is 12 000 and the required return is 12 what is the NPV Would you accept this project

A company is going to make a storage container with sheet steel walls The container will be in the shape of a rectangular prism as shown below If the sheet steel costs 25 for each square meter how much will the sheet steel cost in total 6 m m m 30 X

Lincoln High wants to estimate the number of students who drive to school Answer the following a Which of the following surveys probably would best represent the entire student population 25 students are randomly selected from the 12th grade 6 drive to school 025 students are randomly selected from the school 4 drive to school 025 students are randomly selected from the glee club 3 drive to school b There are 1850 students at Lincoln High Using your answer from part a estimate the number of students who drive to school students

Ten cyclists were each asked the distance in miles they biked last week Their responses are given below Complete the grouped frequency distribution for the data In the distribution the frequency of a class is the number of distances in that class Note that we are using a class width of 5 7 4 Distance in miles 16 17 6 5 12 4 6 10 Distance in miles 1 to 5 6 to 10 11 to 15 16 to 20 Frequency 0 0 0 7 X

Consider a sphere with a radius of 6 1 mm Using the ALEKS deiculator approximate the volume of the sphere To do the approximation do not round any intermediate steps and use the button on the calculator Round your answer to the nearest hundredth If necessary refer to the list of geometry formulas Make sure that you use the correct units in your answer Approximate volume mm mm mm 6 1 mm

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A rectangular carpet has a perimeter of 208 inches The length of the carpet is 84 inches more than the width Find the length and width of the rectangle and then add them together

The table shows fuel consumption in billions of gallons by all non military motor vehicles in selected years What is the quadratic function that models the data f x x Round to three decimal places as needed C Let x 0 correspond to 1970 Using 5 109 0 as the vertex and the dat for 1990 find a quadratic function f x a x h k that models this data the model to estimate fuel consumption in 1998 Year Fuel Consumption 1975 1980 1985 1990 109 0 111 6 118 2 127 5

What is the probability of obtaining six tails when a coin is flipped six times Enter the probability as a fraction

Suppose a pair of dice are rolled Consider the sum of the numbers on the top of the dice and find the probabilities Enter the probabilities as fractions a 10 given that a double was rolled b a double given that an 10 was rolled

What are the odds in favor of drawing a five from an ordinary deck of cards

Suppose that the odds in favor are 9 to 1 that a man will be bald by the time he is 60 State this as a probability Enter the probability as a fraction P a man is bald by 60

Suppose the odds are 35 to 1 that someone will lie to you at least once in the next seven days State this as a probabilit the probability as a fraction

Two cards are drawn from a standard deck of cards and one of the two cards is noted and removed Find the probability of the second card given the information about the removed card provided Enter the probability as a fraction P black red

A single card is drawn from a standard deck of cards Find the probability if the given information is known about the chose Enter the probability as a fraction P club not a spade

Suppose X and Y are random variables with the given joint density function J0 1e 0 5x 0 2y if x 0 y 20 0 1e otherwise f x y a Is f a joint density function Yes No b Find P Y 3 Round your answer to four decimal places 0 5488 Find P X 3 Y 6 Round your answer to four decimal places 0 4287 X c Find the expected value of X Find the expected value of Y

What are the odds in favor of drawing a king from an ordinary deck of cards

Three fair coins are tossed What is the probability that at least one is a tail Enter the probability as a fraction