Student: Sterling White Instructor: David Scott Date: 10/11/22 Course: Math 1324 PreCalculus for Business Assignment: Section 5.3 Homework 8. A manufacturing company makes two types of water skis. a trick ski and a slalom ski. The relevant manufacturing data are given in ti table. Labor-Hours per Ski Maximum Labor-Hours Department Trick Ski m Available per Day Fabricating FinishingB Answer parts (,(A) )and(C )belovv. (A) If the prot on a trick ski Is $60 and the prot on a slalom ski Is $50 how" many of each type of ski should be manufactured each day to realize a maximum profit? What' Is the maximum profit? The maximum profit is '16:. The maximum occurs when :::] trick skis and :: slalom skis are produced. (B) Discuss the effect on the production schedule and the maximum prot if the prot on a slalom ski decreases to $45. The maximum profit (1) W $i::::l. The maximum occurs when :i trick skis and :1 slalom skis are produced. (C) Discuss the effect on the production schedule and the maximum profit if the prot on a slalom ski increases to $55. The maximum prot (2) __ $|:l. The maximum occurs when l:: trick skis and ::| slalom skis are produced. (1) C! decreases to (2) if) increases to Cl increases to {:3 decreases to 0 remains the same at C) remains the same at Student: Sterling White Instructor: David Scott Date: 10/11/22 Course: Math 1324 PreCalculus for Business 21. Assignment: Section 5.3 Homework Because of new federal regulations on pollution, a chemical plant introduced a new, more expensive process to supplement or replac an older process used in the production of a particular chemical. The older process emitted 30 grams of sulfur dioxide and 90 grams particulate matter into the atmosphere for each gallon of chemical produced. The new process emits 5 grams of sulfur dioxide and 3( grams of particulate matter for each gallon produced. The company makes a profit of 80 per gallon and 2052 per gallon on the old ar new processes, respectively. Answer parts (A) through (C) below. (A) If the government allows the plant to emit no more than 25,500 grams of sulfur dioxide and 67,500 grams of particulate matter daily, how many gallons of the chemical should be produced by each process to maximize daily profit? What is the maximum daily prot? Summarize the model. Maximize (1) Objective function subjectto ):| 25,500 Sulfurdioxide constraint (3) :l 67,500 Particulate constraint x, y a 0 Nonnegative constraint Under the stated government regulations a maximum prot of $E::j is obtained by producing :: (4) _ of product using the old process and [:1 (5) _.__._ of product using the new process. (Type integers or decimals. Round to two decimal places as needed.) (B) Discuss the effect on the production schedule and the maximum prot if the government decides to restrict emissions of sulfur dioxide to 17,550 grams daily and all other data remain unchanged. Use a similar process to obtain the constraints for the new mode The maximum obtainable daily profit is now $:] obtained by producing E: (6) _ of product using the old process and E::] (7) ____ of product using the new process. (Type integers or decimals. Round to two decimal places as needed.) (C) Discuss the effect on the production schedule and the maximum profit if the government decides to restrict emissions of sulfur dioxide to 6,450 grams daily and all other data remain unchanged. Under these regulations a maximum profit of $[::::| Es obtained by producing [: (3) __ of product using the old process and [Z (9) __.._..... of product using the new process. (Type integers or decimals. Round to two decimal places as needed.) (1) Q P=5x+30y (2) O at (3) C) a (4) Ct gram(s)pergallon (5) 11:) gallon(s)pergram O P=80x+80y Q 2 C) = C: gallon(s)pergram Ct gram(s) Q P=20x+20y Q s C} s (3 gram(s) Q gram(s) per gallon {:2 P=80x+20y O = C) x {j gallon(s) 13:5 gallon(s) (6) C3 gram(s)pergallon (7) (j gram(s) (8) {:1 gram(s) (9) (:3 gram(s) per gallon Ci gram(s) C} gallon(s) Q gallon(s)pergram IO gallon(s)pergram C2 gallon(s) per gram (:3 gallon(s)per gram (:3 gallon(s) O gallon(s) Ct gallon(s) Ct gram(s)per gallon C) gram(s) per gallon t3 gram(s) Student: Sterling White Instructor: David Scott Date: 10/11/22 Course: Math 1324 PreCalculus for Business Assignment: Section 5.3 Homework 1. Solve the linear programming problem. What is the maximum value of z? Maximize and minimize Select the correct choice below and fill in any answer boxes present in your choice. z= 4x + 5y A. Z= Subject to 2x+y 2 30 "Type an integer or a fraction.) x + 2y 2 24 O B. There is no maximum value of z. X, y 20 What are the coordinates of the corner point where the maximur value of z occurs? Select the correct choice below and fill in any answer boxes present in your choice A. The coordinates are (Type an ordered pair.) B. There is no maximum value of z. What is the minimum value of z? Select the correct choice below and fill in any answer boxes present in your choice. QA. 2= (Type an integer or a fraction.) B. There is no minimum value of z. What are the coordinates of the corner point where the minimum value of z occurs? Select the correct choice below and fill in any answer boxes present in your choice. A. The coordinates are Type an ordered pair.) B. There is no minimum value of z.student: Sterling White Instructor: Davrd Scott Assignment: Section 53 Homework )ate:_'l_g{11__/g2 7 Course: Math 1324 PreCaIculuE for Business\" "Mm\" 7. The corner points for the bounded feasible region determined by J Select the correct choice below and, if necessary, fill in the the system of inequalities shown below are 0 = (0,0), A = (0,5), answer box to complete your choice. B = (3.4), and C = (5,0). ' X + 3y 5 15 , {i} A- The minimum occurs at both 0 and C if a =l 2x+y S 10 x y > 0 O B. There is no value ofa which would cause the minimum in If P = ax +10y find all such numbers a such that the minimum value of P occurs at both 0 and C. dent: Sterling White Instructor: David Scott Date: 10/11/22 Course: Math 1324 PreCalculus for Business Assignment: Section 5.3 Homework 5. Explain why you cannot conclude that optimal solutions exist Why can you not conclude that optimal solutions exist? without graphing the objective function for various feasible solutions. Graph the feasible region and use graphs of the objective function for various values of z to determine the A. Neither a maximum value nor a minimum value exist. maximum value and the minimum value, if they exist. OB. The feasible region is empty and one of the coefficients C. The feasible region is unbounded and one of the coeffici Minimize and maximize O D. The feasible region is bounded and one of the coefficien Z= X -y What are the coordinates of the corner point where the minimum Subject to x - 4y 2 0 value of z occurs? 2x - y 2 14 x, y 2 0 A. There is no minimum. O B. (7,0) O C. (8,2) O D. (0,0) What are the coordinates of the corner point where the maximum value of z occurs? A. There is no maximum. C B. (7,0) O c. (0,0) OD. (8,2) What are the minimum and maximum values of z? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The minimum value of z is . The maximu Type integers or fractions. B. The minimum value of z is | There is no (Type an integer or a fraction. O C. The maximum value of z is . There is no Type an integer or a fraction. O D. There are no minimum nor maximum values of z.Student: Sterling White Instructor: David Scott Date: 10/11/22 Course: Math 1324 PreCalculus for Business Assignment: Section 5.3 Homework 5. Explain why you cannot conclude that optimal solutions exist Why can you not conclude that optimal solutions exist? without graphing the objective function for various feasible solutions. Graph the feasible region and use graphs of the A. Neither a maximum value nor a minimum value exist. objective function for various values of z to determine the maximum value and the minimum value, if they exist. B. The feasible region is empty and one of the coefficients o QC. The feasible region is unbounded and one of the coefficie Minimize and maximize D. The feasible region is bounded and one of the coefficients Z =X-y What are the coordinates of the corner point where the minimum Subject to x - 4y 2 0 value of z occurs? 2x - y 2 14 x, y 2 0 A. There is no minimum. O B. (7,0) C. (8,2) O D. (0,0) What are the coordinates of the corner point where the maximum value of z occurs? O A. There is no maximum. OB. (7,0) O C. (0,0) O D. (8,2) What are the minimum and maximum values of z? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The minimum value of z is . The maximur (Type integers or fractions. B. The minimum value of z is . There is no m (Type an integer or a fraction. C. The maximum value of z is 1. There is nor (Type an integer or a fraction. D. There are no minimum nor maximum values of z.._ DONE The linear programming proolem. Maximize and minimize z=4x+5y Subject to 2x+y z 30 x+2y 2 24 x,y 2 0 Solve the linear programming problem. Maximize P=3Dx+40y Subject to 2x+y s 12 x+y S 7 x+2y 510 x,y 2 0 VVnElI IS The maxrmum value OT Z! Select the correct choice below and fill in any answer boxes present in your choice. ':' A. 2: (Type an integer or a fraction.) {:3 B. There is no maximum value of 2. What are the coordinates of the corner point where the maximurr value of 2 occurs? Select the correct choice below and ll in any answer boxes present in your choice. it? A- The coordinates are (Type an ordered pair.) \":3! B. There is no maximum value of 2. What is the minimum value of 2? Select the correct choice below and fill in any answer boxes present in your choice. {3' A. 2: (Type an integer or a fraction.) Q} B. There is no minimum value of 2. What are the coordinates of the corner point where the minimum value of 2 occurs? Select the correct choice below and ll in any answer boxes present in your choice. [3' A- The coordinates are (Type an ordered pair.) at} B. There is no minimum value 01 2. What is the maximum value of P? Select the correct choice below and ll in any answer boxes present in your choice. " A- P = 240 (Type an integer or a fraction.) 8. There is no maximum value of P. What are the coordinates of the corner point where the maximurr value of P occurs? Select the correct choice below and ll in any answer boxes present in your choice. V A- The coordinates are (4,3) (Type an ordered pair.) B. There is no maximum value of P. 2. Explain why you cannot conclude that optimal solutions exist vvny can you not conclude that optimal solutions exist? without graphing the objective function for various feasible solutions. Graph the feasible region and use graphs of the O A. Neither a maximum value nor a minimum value exist. objective function for various values of z to determine the maximum value and the minimum value, if they exist. O B. The feasible region is empty and one of the coefficients O C. The feasible region is unbounded and one of the coeffic Minimize and maximize O D. The feasible region is bounded and one of the coefficien z= X-y What are the coordinates of the corner point where the minimum Subject to x - 4y 2 0 value of z occurs? 2x - y 2 14 x, y 20 O A. There is no minimum. O B. (7,0) O C. (8,2) O D. (0,0) What are the coordinates of the corner point where the maximum value of z occurs? O A. There is no maximum. O B. (7,0) O C. (0,0) O D. (8,2) What are the minimum and maximum values of z? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The minimum value of z is The maximu Type integers or fractions. O B. The minimum value of z is There is no (Type an integer or a fraction. O C. The maximum value of z is . There is no Type an integer or a fraction. O D. There are no minimum nor maximum values of z. 3. The corner points for the bounded feasible region determined by Which conditions on a and b will ensure that the maximum valu the system of inequalities shown below are O = (0,0), A = (0,9), of P occurs at C? B = (6,6), and C = (8,0). x + 2y S 18 OA. a= 2b 3x+y s 24 x, y 20 If P = ax + by and a, b > 0, determine conditions on a and b that C. a > 2b will ensure that the maximum value of P occurs only at C. a =- 3 G. b > 2a OL. w / a - = Ejs Pm = E: (Type an integer or decimal rounded to the nearest thousandth as needed.) :5. Because Of limited funds, 4 research centers are to be chosen out of 9 suitable ones for a study on neart disease. It the selection Is made at random, what is the probability that 4 particular research centers will be chosen? Let the sample space S be the set of ways 4 research centers that can be chosen out of 9 suitable ones. The number of elements in the sample space S is n(S) = (Type a whole number.) The probability is (Type an integer or a simplified fraction.) 6. Membership selection. A town council has 14 members, 8 Democrats and 6 Republicans. (A) If the president and vice-president are selected at random, what is the probability that they are both Democrats? B) If a 3-person committee is selected at random, what is the probability that Republicans make up the majority? (A) The probability that they are both Democrats is approximately (Type a decimal. Round to three decimal places.) (B) The probability that Republicans make up the majority is approximately (Type a decimal. Round to three decimal places.):1. An experiment consists of rolling two fair ( not weighted) dice and adding the dots on the two sides TacIng up. Each ale nas the numb 1 on two opposite faces, the number 2 on two opposite faces, and the number 3 on two opposite faces. Compute the probability of obtaining the indicated sum. Sum of 4 The probability of getting a sum of 4 is Type an integer or a simplified fraction.) 2. An experiment consists of rolling two fair (not weighted) dice and adding the dots on the two sides facing up. Each die has the numb 1 on two opposite faces, the number 2 on two opposite faces, and the number 3 on two opposite faces. Compute the probability of indicated sum. Sum of 2 The probability of getting a sum of 2 is (Type an integer or a simplified fraction.) 3. Twelve popular brands of beer are used in a blind taste study for consumer recognition. Complete parts (A) and (B) below. (A) If 6 distinct brands are chosen at random from the 12 and if a consumer is not allowed to repeat any answers, what is the probability that all 6 brands could be identified by just guessing? Let the sample space S be the set of ways that the 6 distinct brands can be chosen from the 12 popular brands of beer. The number elements in the sample space S is n(S) = (Type a whole number.) The probability is (Type an integer or a simplified fraction.) (B) If repeats are allowed in the 6 brands chosen at random from the 12 and if a consumer is allowed to repeat answers, what is the probability that all 6 brands are identified correctly by just guessing? The probability is (Type an integer or a simplified fraction.) 4. A laboratory technician is to be tested on identifying blood types from 8 standard classifications. Complete parts (A) and (B) below. (A) If 4 distinct samples are chosen at random from the 8 types and if the technician is not allowed to repeat any answers, what is the probability that all 4 could be correctly identified by just guessing? Let the sample space S be the set of ways that the 4 distinct samples can be chosen from the 8 standard classifications. The number of elements in the sample space S is n(S) = (Type a whole number.) The probability is Type an integer or a simplified fraction.) (B) If repeats are allowed in the 4 blood types chosen at random from the 8 and if the technician is allowed to repeat answers, what the probability that all 4 are identified correctly by just guessing? The probability is (Type an integer or a simplified fraction.)Student: Sterling White Instructor: David Scott Date: 10/12/22 Course: Math 1324 PreCalculus for Business ASSIgnment: Section 8'5 Homework For the following set of numbers, nd the mean. 13, 15,26,11,26,16, 12 The mean is I 17 E. Find the average (mean) of the exam scores 71, 91, 46. 78, and 82, if each score is multiplied by two. The mean of the exam scores when each score is multiplied by 2' Is I 139. 2 (Type an integer or decimal rounded to one decimal place as needed.) EIXI=l::j (Type an integer or a decimal.) i, You draw and keep a single bill from a hat that contains a $1, $10, $20 and $50 bill. What' Is the expected value of the game to you? Let the random variable X represent the Image value of bills. Fill In the probabilities for the probability distribution of the random variabl (Type integers or simplified fractions.) The expected value of the game to you is 95:. (Type an integer or a decimal.) i. You draw and keep a single coin from a bowl that contains 3 pennies. 11 dimes, and 36 quarters. What is the expected value of the game to you? The expected value of the game to you is $. I 0203"l _ .. (Round to three decimal places as needed.) YOU ANSWERED: 20.28 5. You draw a single card from a standard 52-card deck. If it is red, you win $35. Otherwise you get nothing. What is the expected value c the game to you? The expected value of the game to you is $T '. .. I 1_ 7.53 I (Type an integer or a decimal.) ln tossing 2 fair coins, what is the expected number of heads? Let the random variable X represent the number of heads. E(X) =1 I V I 1 I i I (Type an integer or a decimal.) 1. An unfair coin is flipped. IT a nead turns up, you win $1. If a tall turns up, you lose $1. I ne probability of a head is U.42 and the probability of a tail is 0.58. What is the expected value of the game? Let X be the random variable for the amount won on a single play of this game. Fill in the probabilities for the probability distribution of the random variable X $1 X $ - 1 Pi Type integers or decimals.) The expected value of the game is $ (Type an integer or a decimal. Round to the nearest cent as needed.) 1. After paying $3 to play, a single fair die is rolled, and you are paid back the number of dollars corresponding to the number of dots facing up. For example, if a 4 turns up, $4 is returned to you for a net gain, or payoff, of $1, if a 2 turns up, $2 is returned for a net gain of - $1, and so on. What is the expected value of the game? Is the game fair? The expected payoff of the game is $ (Type an integer or a decimal.) Is the game fair? O No Yes 0. Two coins are flipped. You win $3 if either 2 heads or 2 tails turn up, and you lose $1 if a head and a tail turn up. What is the expecte value of the game? The expected value of the game is $ (Type an integer or a decimal.) 1. On three rolls of a single die, you will lose $13 if a 5 turns up at least once, and you will win $6 otherwise. What is the expected value of the game? Let X be the random variable for the amount won on a single play of this game. Fill in the probabilities for the probability distribution of the random variable X -$13 $6 (Type integers or simplified fractions.) The expected value of the game is $ Type an integer or a decimal rounded to the nearest cent as needed.) 2. A pair of fair dice is rolled once. Suppose that you lose $8 if the dice sum to 7 and win $10 if the dice sum to 3 or 2. How much shoul you win or lose if any other number turns up in order for the game to be fair? To keep the game fair, you should (1) $ if the dice sum to any other number. (Do not round until the final answer. Then round to the nearest cent as needed.) (1) O lose win3. A b-cara nand is dealt from a standard 52-card deck. If the b-card nand contains at least one Tive, you win $70; otherwise, you lose $2. What is the expected value of the game? The expected value of the game is | dollars. (Type an integer or a decimal rounded to two decimal places.) 4. The payoff table for two courses of action, A, or A2, is given to the right. Which of the two actions A1 A2 will produce the largest expected value? What is it? Pi X; X; 0.3 . $100 - $400 0.1 $200 $10 0.3 $100 $400 0.3 $100 $100 Action (1) will produce the largest expected value, where the expected value produced is $ Simplify your answer.) (1) O A1 Q A2 5. A game has an expected value to you of $400. It costs $400 to play, but if you win, you receive $100,000 (including your $400 bet) fo a net gain of $99,600. What is the probability of winning? Would you play this game? Discuss the factors that would influence your decision. The probability of winning is (Type an integer or a decimal.) Would you play this game? Discuss the factors that would influence your decision. Since the probability of winning the game is (1) it (2) worth it to play the game. (1) ) very low, (2) @ does not seem 50-50 seems O very high, 6. Five thousand tickets are sold at $1 each for a charity raffle. Tickets are to be drawn at random and monetary prizes awarded as follows: 1 prize of $900, 3 prizes of $200, 5 prizes of $50, and 20 prizes of $5. What is the expected value of this raffle if you buy 1 ticket? Let X be the random variable for the amount won on a single raffle ticket. E(X) = dollars (Round to the nearest cent as needed.) 7. The annual premium for a $10,000 insurance policy against the theft of a painting is $200. If the (empirical) probability that the paintit will be stolen during the year is 0.01, what is your expected return from the insurance company if you take out this insurance? Let X be the random variable for the amount of money received from the insurance company in the given year. E(X) = - 100 dollars8. Decision analysis. After careful testing and analysis, an oil company is considering arilling in two amerent sites. It is estimated Inal s A will net $20 million if successful (probability .3) and lose $2 million if not (probability .7); site B will net $60 million if successful (probability .2) and lose $8 million if not (probability .8). Which site should the company choose according to the expected return from each site? a. What is the expected return for site A? $ million b. What is the expected return for site B? $ million c. Which site should the company choose? Site B O Site A 9. Suppose that at each birth, having a girl is not as likely as having a boy. The probability assignments for the number Number o of boys in a 3-child family are approximated empirically from past records and are given in the table. What is the Boys expected number of boys in a 3-child family? Xi Pi 9 WN - 0.1 The expected number of boys in a 3-child family is Type an integer or a decimal.) A pink-flowering plant is of genotype RW. If two such plants are crossed, we obtain a Number of W Genes Preser red plant (RR) with probability 0.08, a pink plant (RW or WR) with probability 0.86, and a Xi Pi white plant (WW) with probability 0.06, as shown in the table. What is the expected number of W genes present in a crossing of this type? 0.08 0.86 N 0.06 The expected number of white genes (W) is (Type an integer or a decimal.)