Basic Math Questions and Answers

In this problem we prove the following statement Let A B be sets in a universe U Then An Bc C AUB Your task is to provide justification for some of the steps Possible answers are given below Put a number into each blank with no parentheses 1 definition of union 2 definition of intersection 3 definition of subset 4 definition of set subtraction i e S T 5 definition of complement 6 DeMorgan s law of logic Proof Let a A n B Then x E A and x E B by justification Since a A this means that x A by justification Using the same reasoning a E B means that a B so x EB is true Combining these facts A EB is true Therefore EA V EB is true using justification In other words E AUB is true using justification Since AUB this means that a E AUB using justification 9 I So x A is true which is equivalent to x AUB

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle Find the area of the largest possible Norman window with a perimeter of 20 feet Round your answer to one place after the decimal

The graph of 0 5x 2y 3 is shown on the grid 3 2 S 54 142 14 1 3 AT 2 1 14 Z Which ordered pair is in the solution set of 0 5x 2y 3 A 2 0 5 B 2 1 C 2 1 D 2 0 5 4

1 Which ordered pair is in the solution set of y 2x 4 A 6 1 B 1 6 C 6 1 D 1 6 47 wana M in 10 68 1234 S 7 X

oblem 3 Consider the following 5 statements 2 of the statements are false in general Determine which 2 statements are false by testing out each statement on an appropriate matrix like we did with the properties of determinants in Section 3 3 of the tutorial file Note You should not use a magic or pascal matrix for i or ii below because they have special properties not shared by other matrices Try using rand instead 1 i If A is nxn then A and A have the same eigenvalues ii If A is n xn then A and A have the same eigenvectors iii If A is n n then det Ak det A k iv If I is the nx n identity matrix and J is an nxn matrix consisting entirely of ones then the matrix I is invertible and 1 1 J J n 1 n 1 v If I is the nxn identity matrix and J is an n x n matrix consisting entirely of ones then the matrix A I is idempotent i e A A J n 1 Don t forget that you are selecting which statements are false you are not selecting which statements are true A i and iv B iv and v C iii and v D ii and iv E iii and iv F i and v G ii and v H i and ii

4 Listen List the potential rational zeros of the polynomial function f x 2x3 3x 4x 8 0 1 11 12 4 8 O 1 2 4 O 1 2 4 8 10

41 Listen Use the Remainder Theorem to find the remainder when f x Is f x 3x6 5x3 8 x 1 OR 8 R 10 R 6 R 16

4 Listen Form a polynomial f x with real coefficients having the gl Degree 3 zeros 1 i and 5 Of x x x 8x 10 f x x3 5x 8x 12 f x x3 3x 8x 10 x x 3y 108 8

2 Compute gross profit on the sale of Job 201 Complete this question by entering your answers in the tabs below Required 1 Required 2 Prepare a job cost sheet for Job 201 and for Job 202 for the month Materials Labor Overhead Job Number Job Number 201 202

QUESTION 11 4 pts Find the following limits If the limit does not exist explain why You may NOT use guessing by plugging in values a lim 1 0 2e 2x 2 sin x

g 0 6x 5x 6 h 6a 25 25a i 100 16x 60

2 2

In this project you will create your own lottery game with a theme 1 Create at least 5 dollar amount prizes for your lottery game List them from the smallest prize to the largest prize 2 Create the probabilities for each prize in your lottery game using PERCENTS i e 30 0 5 etc Any percent is ok if they add up to 100 You may also want the probability to go down as the prize goes up but it does not have to be 3 Find the expected value for your lo4ery game 4 Using the expected value determine how much you are going to charge for a Ticket to play your game so that you can expect to make money over Eme 5 Choose a theme for your game it can be anything from pirates to sports to superheroes anything goes if it is appropriate 6 Finally create the back of your Lottery Ticket on a separate sheet of paper that shows the prizes and probabilities for your lottery Make sure to also have the cost of your Ticket visible as well Finally decorate the back of the Ticket to fit your theme 7 The Project will be submitted on paper In addition you can also post pictures of your project in Google classroom

Question A boat starts off 172 miles directly west from the city of Johnstown It travels due south at a speed of 30 miles per hour After travelling 126 miles how fast is the distance between the boat and Johnstown changing Do not include units in yo answer and round to the nearest hundredth

Two schools are represented by points A 2 20 and B 14 24 on the graph below A road represented by the line R with equation 2 y 4 passes near the schools An architect is asked to determine the location of a new bus stop on the road such that it is the same distance from the two schools y 28 24 20 16 12 8 4 A 2 20 B 14 24 R

3 km Island Provide your answer below 7 km Tent Shoreline A visitor is staying in a tent that is 7 kilometers east of the closest point on a shoreline to an island The island is 3 kilometers due south of the shoreline The visitor plans to travel from the tent to the island by running and swimming If the visitor runs at a rate of 6 kmph and swims at a rate of 5 kmph how far should the visitor run to minimize the time it takes to reach the island Round to two decimal places

Use the formula above to determine the accumulated amount for each of the following different scenarios Round solutions to the nearest cent Assume there are 365 days in a year If 43 000 is invested for 27 years and earns 12 5 interest compounded annually the accumulated amount is If 43 000 is invested for 27 years and earns 12 5 interest compounded semi annually the accumulated amount is A If 43 000 is invested for 27 years and earns 12 5 interest compounded quarterly the accumulated amount is If 43 000 is invested for 27 years and earns 12 5 interest compounded monthly the accumulated amount is A If 43 000 is invested for 27 years and earns 12 5 interest compounded daily the accumulated amount is

A refrigerator operates at steady state with a coefficient of performance of 4 5 and a power input of 0 8 kW Energy is rejected from the refrigerator to the surroundings at 20 C by heat transfer from metal coils attached to the back Determine a the rate energy is rejected in kW b the lowest theoretical temperature inside the refrigerator in K

Copy the incomplete axes and fill consistent a 0

7 1 11 550 901 481 228 166 420 161 152 119 987 184 147 803 118 2 194 111 5 638 491 S 811 639 11 201 174 14 353 235 200 28 307 83 3 641 441 416 410 27 9 129 110 12 212 184 15 983 676

Two functions g and fare defined in the figure below 8 Domain of g 8 Range of g Domain of f 7 Range of f Find the domain and range of the composition fog Write your answers in set notatic a Domain of fog 0 b Range of fog 0 X 0 0 S

1 point Find a vector a that has the same direction as 10 5 10 but has length 3 Answer a 15 help vectors

assign7 Problem 19 1 point Determine if the pairs of vectors below are parallel orthogonal or neither a 1 4 5 and b 5 20 17 are a 1 4 5 and b 5 20 25 are a 1 4 5 and b 5 20 25 are Note You can earn partial credit on this problem

Find a vector a that has the same direction as 10 5 10 but has length 3

1 point Find the volume of the parallelepiped defined by the vectors Volume 4 ACE 2 2

1 point Find a b if a 10 b 7 and the angle between a and b is 2 Answer

Let X represent the full height of a certain species of tree Assume that X has a normal probability distribution with a mean of 108 6 ft and a standard deviation of 8 2 ft A tree of this type grows in my backyard and it stands 103 7 feet tall Find the probability that the height of a randomly selected tree is as tall as mine or shorter P X 103 7 My neighbor also has a tree of this type growing in her backyard but hers stands 112 7 feet tall Find the probability that the full height of a randomly selected tree is at least as tall as hers P X 112 7

Find the z scores that separates the middle 78 of the distribution from the area in the tail of the standard normal distribution First z score Note If it did not work try switching your answer around Second z score

1 Let s stretch a little on this one Assume Shannon s Irish Red sells 1 000 pints per month in the absence of any cannibalization Assume also that the new Irish Stout will sell 500 pints per month The relevant price and cost data are Price Irish Red 5 00 Unit Variable Costs 2 75 Irish Stout 5 34 3 25 What will be the maximum percentage cannibalization that can exist before the overall change in contribution dollars becomes negative Express your answer in percentage form to the nearest percent e g 99 49 ounds down to 99 99 50 rounds up to 100 Do not include the symbol

ke n values 7 Write as an equation in two variables a d t 3t 5 b f x 6x 4 c C n 5n d P n 2n 7

3 Given this graph of the function f x Find a f 4 e x when f x 2 b f 0 f x f x when f x 0 c f 3 d f 5

5 Swine flu is attacking Porkopolis The function below determines how many people have swine where t time in days and S the number of people in thousands S t 9t 4 a Find S 4 b What does S 4 mean c Find t when S t 23 d What does S t 23 mean e Graph the function

Suppose that the functions u and w are defined as follows u x 2x 2 w x x Find the following u ow 2 wou 2 X S

For each relation decide whether or not it is a function Domain lake tree sun pencil O Function O Not a function Relation 1 Function Not a function Range 9 8 8 Relation 3 u j g j b u j b Domain 8 8 9 6 O Function O Not a function Relation 2 Function Not a function Range 9 4 8 6 4 Relation 4 0 x 5 z 5 m 5 b

Suppose that the functions g and h are defined for all real numbers x as follows g x x 2 h x 4x Write the expressions for h g x and h g x and evaluate h g 3 h g x h g x h g 3

The functions f g and h are defined as follows f x 2x 2 g x 3 x 8 4 x 3 1 x x Find f 4 g 4 and h 4 Simplify your answers as much as possible X

The function h is defined by h x 3x 2 Find h 3x h 3x

The functions f g and h are defined as follows f x 8x 13 g x 2x 2 h x 2 10 Find 2 g 3 and 2 Simplify your answers as much as possible 0 g 3 h 2 0 X S

Homework 14 Distribution Date 10 26 2023 Due Date 10 30 2023 1 Calculate o u o at A B C and D Refer to the figure below H2 H3 Layer 1 H 4 m e 0 5 Gs 2 65 A B C Dry sand Sand Layer 2 H 1 5 m e 0 65 Gs 2 68 Clay Rock Layer 1 Groundwater table Layer 2 Layer 3 Layer 3 H 3 m e 0 81 Gs 2 71 NOTE Show all the calculation steps

Esp The Sugar Sweet Company is going to transport its sugar to market It will cost 6750 to rent trucks and it will cost an additional 250 for each ton of sugar transported Let C represent the total cost in dollars and let S represent the amount of sugar in tons transported Write an equation relating C to S and then graph your equation using the axes below Equation 10 000 C 9000 5000 7000 6000 5000 4000 3000 2000 1000 0 0 10 X 5 X S

Suppose that the height in centimeters of a candle is a linear function of the amount of time in hours it has been purning After 15 hours of burning a candle has a height of 22 centimeters After 29 hours of burning its height is 16 centimeters What is the height of the candle after 19 hours Candle height in cm Burning time in hours centimeters X 3

Drives Co sells portable hard drives They can sell 800 drives when the price is 65 drive and they can sell 920 drives when the price is 35 drive If x represents the number of drives sold determine the following a What is Drive Co s revenue function R x b What is the price per drive in dollars when revenue is maximized c What is the maximum profit in dollars made from the sale of these drives if Drive Co incurs production costs of 205 per drive and has fixed costs of 695

The radioactive element carbon 14 has a half life of 5750 years A scientist determined that the bones from a mastodon had lost 85 5 of their carbon 14 How old were the bones a the time they were discovered The bones were about years old Round to the nearest integer as needed

13 bed bok int rences 3 Complete the below table to calculate the balance sheet data in trend percents with 2019 as base year Note Round your percentage answers to 2 decimal places Assets Current assets Long term investments Plant assets net Total assets 178 806 148 533 123 060 Liabilities and Equity Current liabilities Common stock Other paid in capital Retained earnings Total liabilities and equity KORBIN COMPANY Balance Sheet Data in Trend Percents December 31 2021 2020 and 2019 2021 2020 2019 100 00 100 00 100 00 100 00 100 00 100 00 100 00 100 00 100 00

31 Jake makes a blend of chocolates for the bake sale The dark chocolates are 4 50 per pound and the milk chocolates are 6 00 per pound Jake sells more than 12 pounds total Jake also sells less than 68 50 worth of chocolates Using this information choose the solution set that would be found in the overlapping areas of a graph of Jake s sales O 10 pounds dark 3 pounds milk 11 pounds dark 4 pounds milk 8 pounds dark 3 pounds milk O None of these satisfy both inequalities

ppose that a household s monthly water bill in dollars is a linear function of the amount of water the household uses in hundreds of cubic feet HCF Whe phed the function gives a line with a slope of 1 45 See the figure below he monthly cost for 16 HCF is 46 91 what is the monthly cost for 19 HCF Monthly cost in dollars 46 91 16 Water usage in HCF

2 ABCO 3 A C D EFI G H I 3200 2850 3000 2300 2550 2600 2400 2300 2700 1800 2150 2000 2700 2450 2400 2600 2700 2300 Carolina North ABC Carolina Central DEF Carolina South GHI New Alignment Carolina West ADG Carolina Central BEH Carolina East CFI Part I Critical Thinking of the activity on the previous page 1 In the previous election how many of the 3 seats did each party win Total Votes Total Votes If the new alignment is used and the voting pattern is similar to the previous election how many of the 3 seats will each party win Based upon this difference which party do you believe is proposing the change Attempt to come up with a situation where a type of gerrymandering might occur

AX H

H X

5 Graph the following system on a sheet of paper or graphing utility x 1 and y 3 Determine which of the following points if any satisfy the system More than one answer may be correct check all correct answers 6 6 1 9 5 3 2 9 0 3 4 6 0 0 None of these points satisfy the system