MATH 1324 Review for Test 3 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the value(s) of the function on the given feasible region. 1) Find the maximum and minimum of z = 20x + 4y. 1) Use graphical methods to solve the linear programming problem. 2) Maximize z = 6x + 7y subject to: 2x + 3y 12 2x + y 8 x 0 y 0 2) Find the value(s) of the function, subject to the system of inequalities. 3) Find the maximum and minimum of P = 19x + 3y subject to: 0 x 10, 0 y 5, 3x + 2y 6. 3) State the linear programming problem in mathematical terms, identifying the objective function and the constraints. 4) A firm makes products A and B. Product A takes 2 hours each on machine L and machine 4) M; product B takes 4 hours on L and 2 hours on M. Machine L can be used for 8 hours and M for 7 hours. Profit on product A is $6 and $7 on B. Maximize profit. Provide an appropriate response. 5) To determine the shading when graphing 7x + 4y 0, the point (0,0) would make a good test point. True or false? 5) 1
Convert the constraints into linear equations by using slack variables. 6) Maximize z = 2x1 + 8x2 Subject to: x1 + 6x2 15 7x1 + 5x2 25 x1 0, x2 0 6) Introduce slack variables as necessary and write the initial simplex tableau for the problem. 7) Maximize z = 4x1 + x2 subject to: 2x1 + 5x2 11 3x1 + 3x2 19 x1 0, x2 0 7) Find the pivot in the tableau. 8) 8) Use the indicated entry as the pivot and perform the pivoting once. 9) 9) Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to 0. 10) x1 x2 x3 x4 x5 z 10) 3 4 0 3 1 0 18 1 5 1 7 0 0 25-3 4 0 1 0 1 19 2
A bakery makes sweet rolls and donuts. A batch of sweet rolls requires 3 lb of flour, 1 dozen eggs, and 2 lb of sugar. A batch of donuts requires 5 lb of flour, 3 dozen eggs, and 2 lb of sugar. Set up an initial simplex tableau to maximize profit. 11) The bakery has 580 lb of flour, 660 dozen eggs, 700 lb of sugar. The profit on a batch of 11) sweet rolls is $93.00 and on a batch of donuts is $62.00. A manufacturing company wants to maximize profits on products A, B, and C. The profit margin is $3 for A, $6 for B, and $15 for C. The production requirements and departmental capacities are as follows: Department Production requirement by product (hours) Departmental capacity (Total hours) A B C Assembling 2 3 2 30,000 Painting 1 2 2 38,000 Finishing 2 3 1 28,000 12) What is the constraint for the assembling department? 12) 13) What are the coefficients of the objective function? 13) Tell whether the statement is true or false. 14) 8 {16, 24, 32, 40, 48} 14) Insert " " or " " in the blank to make the statement true. 15) {2, 4, 6} {1, 2, 3, 4, 6} 15) 16) {7, 21, 26} {4, 21, 26, 36} 16) Find the number of subsets of the set. 17) {12, 13, 14} 17) Find the complement of the set. 18) {x x is an integer strictly between 0 and 10} if U is the set of all integers 18) Decide whether the statement is true or false. 19) {9, 18, 27, 36} {9, 27} = {9, 18, 27, 36} 19) Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. 20) B' 20) 21) A B' 21) 22) (A B)' 22) 3
Let U = {all soda pops}; A = {all diet soda pops}; B = {all cola soda pops}; C = {all soda pops in cans}; and D = {all caffeine-free soda pops}. Describe the given set in words. 23) A B 23) Shade the Venn diagram to represent the set. 24) A' B' 24) Use a Venn Diagram and the given information to determine the number of elements in the indicated set. 25) n(u) = 60, n(a) = 35, n(b) = 24, and n(a B) = 3. Find n(a B)'. 25) 26) n(a) = 75, n(b) = 83, n(c) = 77, n(a B) = 15, n(a C) = 17, n(b C) = 11, n(a B C) = 9, and n(a' B' C') = 151. Find n(u) 26) Write the sample space for the given experiment. 27) A box contains 10 red cards numbered 1 through 10. One card is drawn at random. 27) 28) A box contains 2 blue cards numbered 1 through 2, and 3 green cards numbered 1 through 3. A blue card is picked, followed by a green card. 28) Determine whether the given events are disjoint. 29) Drawing a spade from a deck of cards and drawing an ace 29) For the experiment described, write the indicated event in set notation. 30) A die is tossed twice with the tosses recorded as an ordered pair. Represent the following event as a subset of the sample space: The second toss shows a four. 30) A die is rolled twice. Write the indicated event in set notation. 31) The sum of the rolls is 8. 31) Find the probability of the given event. 32) A card drawn from a well-shuffled deck of 52 cards is a red ace. 32) 33) A bag contains 4 red marbles, 5 blue marbles, and 3 green marbles. A randomly drawn marble is blue. 33) 4
Use the given table to find the probability of the indicated event. Round your answer to the nearest thousandth. 34) College students were given three choices of pizza toppings and asked to choose one 34) favorite. The following table shows the results. toppings freshman sophomore junior senior cheese 13 10 24 25 meat 29 25 10 13 veggie 10 13 29 25 A randomly selected student prefers a cheese topping. An experiment is conducted for which the sample space is S = {a, b, c, d}. Decide if the given probability assignment is possible for this experiment. 35) 35) Outcomes Probabilities a.47 b.41 c.12 d.20 Solve the problem. 36) One card is selected from a deck of cards. Find the probability of selecting a red card or a queen. 36) Suppose P(C) =.048, P(M C) =.044, and P(M C) =.524. Find the indicated probability. 37) P(M) 37) 38) P[(M C)'] 38) Convert the odds that the given event will occur to the probability that the event will occur. 39) The odds in favor of winning a particular lottery are 1 to 2,100,000. 39) Find the odds in favor of the indicated event. 40) Spinning an A on the spinner pictured below. (The sectors are of equal size.) 40) Solve the problem. 41) A survey revealed that 50% of people are entertained by reading books, 33% are entertained by watching TV, and 17% are entertained by both books and TV. What is the probability that a person will be entertained by either books or TV? Express the answer as a percentage. 41) 5
42) Below is a table of data from a high school survey given to 500 parents. Find the probability that a randomly chosen parent would agree or strongly agree that the school is clean. Round your answer to the nearest hundredth. Strongly Disagree Disagree Neutral Agree Strongly Agree The school is safe 88 132 100 112 68 The school is clean 73 152 100 127 48 42) 43) If a single fair die is rolled, find the probability of a 5 given that the number rolled is odd. 43) 44) If two cards are drawn without replacement from an ordinary deck, find the probability that the second card is a face card, given that the first card was a queen. 44) Assume that two marbles are drawn without replacement from a box with 1 blue, 3 white, 2 green, and 2 red marbles. Find the probability of the indicated result. 45) The second marble is red, given that the first marble is white. 45) 46) The second marble is blue, given that the first marble is red. 46) 47) Both marbles are white. 47) Decide whether the two events listed are independent. 48) Two cards are selected, without replacement, from an ordinary deck. F is the event that an ace appears on the first draw. S is the event that an ace appears on the second draw. 48) Find the probability. 49) If 81% of scheduled flights actually take place and cancellations are independent events, what is the probability that 3 separate flights will take place? 49) Use the given table to find the indicated probability. 50) College students were given three choices of pizza toppings and asked to choose one favorite. The following table shows the results. 50) toppings freshman sophomore junior senior cheese 15 13 22 23 meat 29 23 13 15 veggie 13 15 29 23 P(favorite topping is meat student is junior)? Round the answer to the nearest hundredth. 6
51) The following table contains data from a study of two airlines which fly to Smalltown, USA. 51) Number of flights arrived on time Number of flights arrived late Podunk Airlines 33 6 Upstate Airlines 43 5 P(flight was on Upstate Airlines flight arrived late)? Prepare a probability distribution for the experiment. Let x represent the random variable, and let P represent the probability. 52) Two balls are drawn from a bag in which there are 4 red balls and 2 blue balls. The number 52) of blue balls is counted. 53) Three cards are drawn from a deck. The number of kings is counted. 53) Give the probability distribution and sketch the histogram. 54) A class of 44 students took a 10-point quiz. The frequency of scores is given in the table. Number of Points Frequency 5 2 6 5 7 10 8 15 9 9 10 3 Total: 44 54) 7
Answer Key Testname: MATH 1324 REVIEW FOR TEST 3 1) 220, 12 2) Maximum of 32 when x = 3 and y = 2 3) 205, 9 4) Maximize 6A + 7B Subject to: 2A + 4B 8 2A + 2B 7 A, B 0. 5) False 6) x1 + 6x2 + s1 = 15 7x1 + 5x2 + s2 = 25 7) x1 x2 s1 s2 z 2 5 1 0 0 11 3 3 0 1 0 19-4 -1 0 0 1 0 8) 4 in row 2, column 2 9) 10) x1 = 0, x2 = 0, x3 = 25, x4 = 0, x5 = 18, z = 19 11) x1 x2 s1 s2 s3 s4 3 5 1 0 0 0 580 1 3 0 1 0 0 660 2 2 0 0 1 0 700-93 - 62 0 0 0 1 0 12) 2A + 3B + 2C 30,000 13) 3, 6, 15 14) False 15) 16) 17) 8 18) {x x is an integer 0 or 10} 19) False 20) {r, t, u, v, w, x} 21) {u, w} 22) {r, t, v, x} 23) All diet-cola soda pops 24) 8
Answer Key Testname: MATH 1324 REVIEW FOR TEST 3 25) 4 26) 352 27) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 28) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} 29) No 30) {(1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4)} 31) {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)} 1 32) 26 33) 5 12 34).319 35) No 7 36) 13 37).520 38).476 1 39) 2,100,001 40) 3 to 5 41) 66% 42).35 43) 1 3 44) 11 51 45) 2 7 46) 1 7 47) 3 28 48) No 49).53 50).203 5 51) 11 52) x P 0.4 1.53 2.07 9
Answer Key Testname: MATH 1324 REVIEW FOR TEST 3 53) 54) x P 0.7826 1.2042 2.0130 3.0002 Number 5 6 7 8 9 10 Probability.05.11.23.34.20.07 10