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Consider the linear programming problem: Maximize ð¶ = 4ð‘¥ + 5𑦠Subject to { ð‘¥ + 2𑦠 10  ð‘¥ + 𑦠 2 4ð‘¥ + 𑦠 20 ð‘¥, 𑦠 0 Graph the feasible set, labelling all lines and corner points. Then find the maximum value ð¶ can obtain. Transcribe the following problem into a linear programming problem by stating the objective equation, along with all constraining inequalities. You do not need to solve it! A team produces two types of doghouses: large and small. Each large house requires 3 hours to construct, 2 hours to paint and 0.5 hours for testing. Each small house requires 2 hours to construct, 1 hour to paint and 0.5 hours for testing. The large houses earn a profit of $100, while the small houses earn a profit of $70. There are 22 hours available for building, 14 hours available for painting and 4.5 hours available for testing. What is the maximum profit the team can obtain? Express the shaded region in the Venn diagram below as some union, intersection and/or complement of the sets ð´, ðµ and ð¶. A doghouse manufacturing team has produced 540 doghouses. Of these: 100 houses have air conditioning 160 houses have plumbing 290 houses have beds 60 houses have both air conditioning and plumbing 130 houses have beds, but not air conditioning nor plumbing 130 houses have both plumbing and beds 210 houses have beds, but not air conditioning Let ð´, ð‘ƒ, ðµ denote the sets of houses with air conditioning, plumbing and beds. Build a 3-circle Venn diagram and fill in all regions with the number of houses contained within that region. Then give the number of houses that have none of the three features. Let 𑈠= {1, 2, 3, 4, 5, 6, 7, 8} be the universal set with ð´ = {1, 2, 3} and ðµ = {3, 4, 5} and ð¶ = {2, 4, 6}. Find ð´â€² ∩ (ðµ ∪ ð¶). How many four-letter words exist (including nonsense words) that satisfy all of the following conditions: No letter is repeated. The first letter is a vowel (A, E, I, O or U). The last letter is not a vowel. An accounting division has ten accountants. Some are to be divided into groups to aide two clients: Clients A and B. How many ways can they be selected so that Client A receives four accountants and Client B receives three (some accountants will be left unassigned)? Tom and Jerry are two friends in the same class of nine people (including them). There are six tables in the classroom, each containing two seats (similar to the classroom used in our lectures, but with only six tables). How many seating arrangements exist so that Tom and Jerry sit at the same desk? Mr. Tinsley plays nine checkers games against the computer program Chinook, and the sequence of wins, losses or ties (draws) is recorded. How many sequences are possible if Mr. Tinsley won at least one game? A teacher has 10 gold stars and 9 smiley face stickers to give to six students. Suppose the stickers are selected randomly. What is the probability that at least four students will get a smiley face? A teacher assigns their students a book report. There are ten different books to choose from. There are seven students in the class, and each student selects a book at random. What is the probability that at least two students will choose the same book? Suppose you are flipping a coin 100 times. Miraculously, the first 99 flips have all been tails. What is the probability that the final flip, and thus all 100 flips, will be tails? Suppose a table illustrating the number of pink and white cherry blossoms in Victoria and Saanich is given in the table below. Using the table, find the probability that a sampled pink cherry blossom came from Victoria. Pink White Total Victoria Saanich Total. A doghouse distributor has both an online and an ‘offline’ retail store. They find that when a person shops online, they have a 72% probability of purchasing a deluxe model, whereas offline retail shoppers have a 54% probability of purchasing a deluxe model. 80% of all transactions with the distributor are online. Build a tree-diagram illustrating this scenario and labelling all events clearly. If a person did not purchase a deluxe model, what is the probability that it was purchased online? A fair game has two outcomes: win $60 or lose $100. What is the probability of winning? A promotor invites 200 people to a party. If each person has a 10% chance of attending the event, what is the probability that exactly twelve people will attend the event? After performing the Gauss-Jordan Elimination Method on a system of linear equations (with variables ð‘¥, ð‘¦, ð‘§, 𑤠as usual), the resulting augmented matrix remains. Write the solution to the original system. Solve the given system of linear equations using matrix inverses. No credit will be given for any other method. { 4ð‘¥ + 6𑦠= 2 3ð‘¥ + 5𑦠= . Find the inverse of the matrix below using the Gauss-Jordan Elimination Method. Show all work. Suppose Lego begins tracking consumers’ preferences between Lego brand toys and their major competitor – Mega Bloks. They find that each year, 15% of consumers that preferred Lego switch to preferring Mega Bloks. Also, 20% of consumers that preferred Mega Bloks switch to preferring Lego each year. If half of the current population prefers Lego, find the proportion that will prefer Lego two years from now. Use Markov chains in your work. In the scenario above, what proportion will prefer Lego in the long run? ð´ = ð‘ƒ(1 + ð‘Ÿð‘¡) ð´ = 𑃠(1 + ð‘Ÿ ð‘š ) ð‘šð‘¡ ð´ = ð‘ƒð‘’ð‘Ÿð‘¡ ð¹ð‘‰ = ð‘ƒð‘€ð‘‡ In your work, show which of the above formulas is being used to solve this. If $1000 is deposited in an account with an annual interest rate of 2%, compounded quarterly, what will be the value of the account after 5 years? In your work, show which of the above formulas is used to solve this. Garret opens an RRSP account with an interest rate of 5% compounded monthly. He pays $400 each month into the account. How much will the RRSP be worth in 35 years? Clearly indicate which formula is being used. The current nominal interest rate on Canada Student Loans is 5.95%, compounded monthly. If a student takes out a $20,000 student loan, and wishes to make payments each month for exactly 5 years, how much will each payment be? Clearly indicate which formula is being used. Consider the problem above. How much of the amount paid on the loan be due to interest? Given the statements below, write (~ð‘ ∧ ~ð‘ž) → ð‘Ÿ in conversational English: ð‘ = "The budget is balanced. " ð‘ž = "Taxes will be raised. " ð‘Ÿ = "Programs will be cut. Let ð‘ be the statement: “The ferry will depart at 9am.†Let ð‘ž be the statement, “The ferry will arrive at noon.†Write the following statements in logical syntax using ð‘, ð‘ž and connectives. “The ferry will not depart at 9am, but will arrive at noon anyway.†Construct a truth table for the statement ð‘ → (~𑞠∨ ð‘).

Paper For Above instruction

The following comprehensive analysis addresses the various mathematical and logical problems presented, demonstrating thorough work and detailed reasoning for each. The solutions span multiple disciplines, including linear programming, set theory, combinatorics, probability, matrix algebra, Markov chains, financial mathematics, and logical statements.

Linear Programming and Graphical Solution

Given the problem of maximizing ð¶ = 4ð‘¥ + 5𑦠with the constraints ð‘¥ + 2𑦠≤ 10, -ð‘¥ + 𑦠≤ 2, 4ð‘¥ + 𑦠≤ 20, and non-negativity conditions ð‘¥, 𑦠≥ 0, I first graph the feasible region. The lines are plotted for each constraint:

• Line 1: ð‘¥ + 2𑦠= 10
• Line 2: -ð‘¥ + 𑦠= 2
• Line 3: 4ð‘¥ + 𑦠= 20

Intersections of these lines identify corner points. Solving these equations pairwise yields the corner points:

• Intersection of Line 1 and Line 2: solving equations, ð‘¥ + 2ð‘¦=10 and -ð‘¥ + ð‘¦=2:
• Adding the equations appropriately, solutions are found at (2,4).
• Intersection of Line 1 and Line 3: ð‘¥ + 2ð‘¦=10 and 4ð‘¥ + ð‘¦=20, solving yields (0,5).
• Intersection of Line 2 and Line 3: solutions give (2,4. ]

  The feasible region is the polygon bounded by these points, and the maximum of ð¶ is found at one of these vertices. Evaluating ð¶ at these points shows maximum at (2,4) with ð¶=4(2)+5(4)=8+20=28. Thus, maximum value of ð¶ is 28.

  Linear Programming Transcription

  Variables: number of large houses (x) and small houses (y). Objective: Maximize Profit = 100x + 70y. Constraints:

  • 3x + 2y ≤ 22 (construction hours)
  • 2x + y ≤ 14 (painting hours)
  • 0.5x + 0.5y ≤ 4.5 (testing hours)
  • x,y ≥ 0

  This LP formulation captures the problem accurately without solving it.

  Set Theory and Venn Diagrams

  Examining the sets ð´, ðµ, ð¶ relative to the given problem, the shaded region involves unions, intersections, and complements. For the set relationships involving the three sets, the union ð´ ∪ ðµ ∪ ð¶ represents all houses with any feature, while their intersections identify houses with multiple features. The complement of the union indicates houses with no features.

  Venn Diagram for Doghouses

  The data indicates:

  • Total houses: 540
  • With A/C: 100
  • With Plumbing: 160
  • With Beds: 290
  • Both A/C and Plumbing: 60
  • With Beds only, nor A/C nor Plumbing: 130 (given)
  • Both Plumbing and Beds: 130
  • Bed only: 210

  Using inclusion-exclusion principles, the counts for each region are computed, with the region having none of these features found by subtracting the sum of all identified houses from the total. The calculated none-feature houses are 40.

  Set Operations and Cardinalities

  Given 𑈠= {1, 2, 3, 4, 5, 6, 7, 8}, ð´ = {1, 2, 3}, ðµ = {3, 4, 5}, and ð¶ = {2, 4, 6}, the operation ð´' ∩ (ðµ ∪ ð¶) involves complements and unions. Since ð´' = {4, 5, 6, 7, 8}, and ðµ ∪ ð¶} = {2, 3, 4, 5, 6}, their intersection yields {4, 5, 6}.

  Counting Four-Letter Words

  Number of words satisfying constraints: no repeated letters, first letter vowel (A, E, I, O, U), last letter not vowel. For 5 vowels, first position options = 5. Last position options = 21 (excluding vowels). Remaining letters are selected from the remaining 24 letters, avoiding repetitions, in 24×23×22 ways. Total words = 5 × 24×23×22×21 = 5×255024 = 1,275,120.

  Accounting Group Selection

  The number of ways to select 4 accountants for Client A: C(10,4)=210. Then, select 3 from remaining 6 for Client B: C(6,3)=20. Remaining accountants are unassigned, so total ways = 210×20=4200.

  Seating Arrangements for Tom and Jerry

  Number of arrangements: they sit together at same table. Number of pairs of seats at each of 6 tables (2 seats each): 6. For each table, Tom and Jerry can sit in 2! ways: 2. Total arrangements: 6×2=12.

  Game Outcomes with Record Sequences

  Number of sequences in 9 games where Mr. Tinsley wins at least once: total sequences=3^9=19683. Sequences with no wins: T, L, or D, all negative, total=2^9=512 (since only losses or draws). Therefore, sequences with at least one win=19683−512=19171.

  Probability of Smiley Stickers

  Total stickers: 10 stars + 9 smileys =19. Each of the 6 students receives 1 sticker randomly. Probability that at least 4 students get smiley faces involves calculating combinations where 4, 5, or 6 students get smileys, divided by total arrangements. Using binomial probabilities, P=∑_{k=4}^6 C(6,k) (9/19)^k (10/19)^{6−k}.

  Probability of Student Book Choices

  For 7 students choosing from 10 books, the probability that at least two select the same book:

  Total outcomes: 10^7.

  Number of outcomes with all different books: P(10,7)=10×9×8×7×6×5×4=604800.

  Probability that all are different: 604800/10^7.

  Therefore, probability of at least one duplicate: 1−(604800/10^7)=approx. 0.94.

  Coin Flips

  Probability that the final (100th) flip is tails given the first 99 are tails: Since flips are independent, ½.

  Pink Cherry Blossom Sampling

  Assuming the table contains counts, probability that a sampled pink blossom from Victoria = (pink blossoms in Victoria) / (total pink blossoms). Specific counts are needed for calculation, but the method is straightforward.

  Tree Diagram for Consumer Purchases

  Construct a tree with branches: online (probability 0.8) and offline (0.2). Each branch divides into: purchase deluxe (72% online, 54% offline) or not. Conditional probability that the purchase was online given no deluxe: by Bayes theorem, P(online | no deluxe)=P(no deluxe | online)*P(online)/P(no deluxe).

  Win/Loss Probability

  Probability of winning: Since only two outcomes, the probability is determined by game rules. If 'win' occurs with probability p, then total expected value is p×60+(1−p)(−100). For a fair game, P(win)= as per balancing the expected values.

  Invite Attendance Probability

  Number of attendees X∼Binomial(n=200,p=0.1). Probability exactly 12 attend: C(200,12)×(0.1)^12×(0.9)^{188}.

  Linear System Solutions

  Gauss-Jordan elimination yields solutions for the systems. For example, the augmented matrix (see problem) can be expressed as x=..., y=..., etc., once simplified.

  Using matrix inverses for system { 4x + 6y=2, 3x + 5y=...} involves finding the inverse of coefficient matrix and multiplying by constants.

  Matrix Inversion via Gauss-Jordan

  The inverse of the matrix:

  |1 0 2|
  |0 1 -1|
  

  is computed through Gauss-Jordan process: augment with identity, row operations produce the inverse matrix.

  Markov Chain Population Preferences

  Model the preferences with transition matrix:

  	To Lego	To Mega Bloks
  From Lego	0.85	0.15
  From Mega Bloks	0.20