Okanagan College Math 111 (071) Fall 2010 Assignment Three Problems & Solutions

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1 Okanagan College Math 111 (071) Problems & s Instructor: Clint Lee Due: Thursday, December 9 Instructions. Do all parts of all 10 questions. Show all significant steps in your work. You will not receive any marks for just an answer, even if it is correct. Use complete English language sentences to answer questions where a written answer or explanation is required. The number of points for each question is given in the left margin, total 80. The questions in this assignment consist of all of the questions on Term Test Three plus additional questions. The questions that appeared on Term Test Three are labeled with a number in square brackets, giving the corresponding number of the question on the test Let A Compute which ever of AB or BA can be formed. & B Since A is 3 3 and B is 2 3 only BA can be formed. The product is BA Sal s Shoes and Fred s Footwear both have outlets in California and Arizona. Sal s sells shoes for $80, sandals for $40, and boots for $120. Fred s prices are $60, $30, and $10 for shoes, sandals, and boots, respectively. During a certain week the sales in California were 00 shoes, 20 sandals, and 20 are boots. During the same week in Arizona the sales were 200 shoes, 200 sandals, and 600 boots. Let P be the 2 3 matrix giving the prices for the two stores and the three types of footwear, where P Shoes Sandals Boots Sal s Fred s (a) Write a 3 2 matrix called F giving the sales for each type of footwear in each state. The matrix F is F Cal Ariz Shoes Sandals Boots Problem 2 continued...

2 Problems & s Page 2... Problem 2 continued 2 (b) Explain why both PF and FP can be formed. The matrix P is 2 3 and the matrix F is 3 2. Hence, both matrix products can be formed since for PF the number of columns on the left is 3 which equal to the number of rows on the right, and for FP the number of columns on the left is 2 which equal to the number of rows on the right. 3 (c) Explain why only the product PF is meaningful. Calculate the product, and describe what the entries represent. Only PF is meaningful, since in this case the column labels on the left match the row labels on the right. Both refer to the footwear type. Each term in the sum of products giving the matrix product is the price for a type of footwear times the number of that type of footwear sold in each store. The product is PF Cal Ariz Sal s Fred s The entries in the matrix product represent the total sales revenues at each store in each state. 3. Consider the matrix A (a) 2a Verify that B is the inverse of the matrix A by computing the matrix product BA. The product is BA Since the result of the multiplication is the identity matrix, BA I, the matrix B is the inverse of matrix A. That is, A 1 B. Problem 3 continued...

3 Problems & s Page 3... Problem 3 continued 2 (b) 2b Use the inverse from part (a) to solve the system of linear equations 3x 1 x 2 + x 3 3 x 1 + x 2 3 x 1 + x 3 2 The system can be written as AX Y where A is the coefficient matrix of the system, which is the matrix A given above, X is the column matrix of unknowns, and Y is the column matrix of right-hand sides in the system, so that Then the solution is given by X A 1 Y BY X x y z So the solution is x 2, y, and z 4. and Y (c) Use the inverse from part (a) to solve the system of linear equations x 1 + x 2 x 3 2 x 1 + 2x 2 x 3 1 x 1 x 2 + 2x The system can be written as BX Y where B is the coefficient matrix of the system, which is the matrix B given above, X is the column matrix of unknowns, and Y is the column matrix of right-hand sides in the system, so that X x y z and Y Since A is the inverse of the matrix B, the solution to the system is given by X B 1 Y AY So the solution is x 3, y 3, and z

4 Math 111 (071) Problems & s Page 4 4. Consider the system of linear inequalities 2x + 3y 30 2x + y 26 2x + y 34 x 0, y 0 4 (a) 3a Draw the feasible set for the system of linear inequalities above, and determine its vertices. The boundary lines are 2x + 3y 30 (line I), 2x + y 26 (line II), and 2x + y 34 (line III). For line I the x and y-intercepts are (1, 0) and (0, 10), respectively. For line II the x-intercept is (13, 0) and another point on the line is (, 16). For line III two points on the line are (3, 8) and (13, 12). y C(3, 8) B(8, 10) A(12, 2) III 2 (b) 3b Maximize the objective function P 20x + 10y subject to the constraints in the given system of linear inequalities. II x I Evaluating P at the three vertices gives A(12, 2) :P 20(12) + 10(2) 260 B(8, 10) :P 20(8) + 10(10) 260 C(3, 8) :P 20(3) + 10(8) 140 So P is maximum for either x 12 and y 2 or x 8 and y 10. In fact, P is maximum at any point on the line joining vertices A and B. 1 (c) 3c Minimize the objective function P 20x + 10y subject to the constraints in the given system of linear inequalities. From the results above it we see that the minimum takes place at vertex C where x 3 and y 8.. Let U { all employees of Okanagan College } M { all mathematics professors at OC } E { all English professors at OC } F { all female professors at OC } Describe each set using an English language phrase. 2 (a) 4a M F 2 (b) 4b M E This is the set of all mathematics professors at OC who are not female, that is, all of the male mathematics professors at OC. This is the set of all employees of OC who are neither mathematics professors nor English professors.

5 Problems & s Page 6. In this problem the universal set is the set of letters U { a, b, c, d, e, i, o, p, q, r, s, t, u, v, x, z }. Let A be the set of letters in the word asterix, let V be the set of vowels, and let Q be the set of letters in the word quitter. 1 (a) a List the elements in the set Q. Q { e, i, q, r, t, u } 2 (b) b Determine A V and describe it in words. The intersection of the sets A and V is the set of letters in common between the two sets. It is A V { a, e, i } This is the set of vowels in the word asterix. 3 (c) c Use set theoretic notation to express the set consisting of the letters that are vowels or are in the word quitter. List the elements in this set. This is the union of the sets Q and V. It is Q V { a, e, i, o, q, r, t, u } 2 (d) Determine A and describe it in words. The complement of set A is the set of letters in the universal set U that are not in A. This is A { b, c, d, o, p, q, u, v, z } This is the set of letters in U that are not in the word asterix. 2 (e) Determine (A Q) and describe it in words. The set A Q is the set of letters either in the word asterix or in the word quitter, or both. So that A Q { a, e, i, q, r, s, t, u, x } Its complement is the set of letters in U that are neither in the word asterix nor in the word quitter. So that (A Q) { b, c, d, o, p, v, z } Problem 6 continued...

6 Problems & s Page 6... Problem 6 continued 3 (f) Verify that (A Q) A Q using the sets A, Q, and U defined in this problem. Draw a Venn diagram to demonstrate that this is true in general. From part (d) A { b, c, d, o, p, q, u, v, z } and Q { a, b, c, d, o, p, s, v, x, z }, so the intersection is A Q { b, c, d, o, p, v, z } (A Q) This is the same set as that in part (e) above. A two set Venn diagram for (A Q) is shown on the left below as the portion of the diagram that is hatched with the diagonal lines. It is everything outside of either A or B. A two set Venn diagram for A Q is shown on the right as the portion of the diagram that is hatched in both directions. It is everything that is outside A and outside B. The two regions are the same. A Q A Q (A Q) A Q Problem 6 continued...

7 Problems & s Page 7... Problem 6 continued 2 (g) Verify that n (A Q) n (A) + n (Q) n (A Q) for the sets A and Q defined in this problem. Just counting the elements in each set we have n (A Q) 9, n (A) 7, n (Q) 6, and n (A Q) 4. Thus, n (A) + n (Q) n (A Q) n (A Q) 7. Suppose that out of 800 first-year students at a certain community college, 30 are taking English, 300 are taking mathematics, and 270 are taking both English and mathematics. 2 (a) 6a How many students are taking either English or mathematics? Let U be the set of all of the first-year students at the community college. This is the universal set. Let E be the set of students taking English and M the set of students taking mathematics. Then the given information is n (U) 800, n (E) 30, n (M) 300, and (E M) 270 The set E M is the set of students taking either English or mathematics. Using the inclusionexclusion principle gives n (E M) n (E) + n (M) n (E M) (b) 6b How many students are taking neither English nor mathematics? The set (E M) E M is the set of students taking neither English nor mathematics. The number of students in this set is n (E M) n (U) n (E M) (c) How many students are taking either English or mathematics, but not both? The set (E M) (E M) is the set of students taking either English or mathematic, but not both. The number of students in this set is n (E M) (E M) n (E M) n (E M)

8 Problems & s Page 8 8. A supplier for the automobile industry manufactures car and truck frames at two different plants. The production rates (in frames per hour) for each plant are given in the table: Plant A Plant B Car frames 10 8 Truck frames 8 The supplier has two orders given in the following table. Order 1 Order 2 Car frames Truck frames (a) Let x be the number of hours worked on Order 1 at Plant A and y the number of hours worked on Order 2 at Plant B. Set up a system of linear equations for the variables x and y that gives the number of hours each plant should be scheduled to operate in order to fill Order 1 exactly. From the first table above { 10x + 8y 3000 x + 8y (b) Find the inverse of the coefficient matrix of the system of equations in part (a). The coefficient matrix is A The determinate is , so the inverse is A (c) Use the inverse from part (b) to find the number of hours each plant should be scheduled to operate to exactly fill Order 1. The system is AX Y, where A is the coefficient matrix above and x 3000 X and Y y 1600 The solution to the system is X A 1 Y So they should have Plant A work on Order 1 for 280 hours and Plant B for 2 hours. Problem 8 continued...

9 Problems & s Page 9... Problem 8 continued 1 (d) Use the inverse again to find the number of hours each plant should be scheduled to operate to exactly fill Order 2. Now and the solution is X A 1 Y Y So they should have Plant A work on Order 1 for 160 hours and Plant B for 10 hours. 9. A machine shop manufactures two types of bolts, lag bolts and carriage bolts. The bolts require time of each of three groups of machines, but the time required on each group differs, as shown in the table below. Lag Carriage Group min 0.2 min Group min 0.2 min Group min 0.08 min Production schedules are made up one day at a time. On a certain day, 300, 720, and 100 minutes are available for each group of machines, respectively. Lag bolts sell for 1 and carriage bolts for 20. It is desired to set up the schedule so that the total return is as large as possible. 2 (a) 7a Define appropriate variables for this linear programming problem and write the objective function in terms of these variables. State whether the objective function is be to maximized or minimized. Let x be the number of lag bolts produced and y be the number of carriage bolts. The objective function is the return from producing x lag bolts and y carriage bolts. This is R 1x + 20y and it is desired to maximize this objective function. Problem 9 continued...

10 Math 111 (071) Problems & s Page Problem 9 continued 4 (b) 7b Give the constraints for this linear programming and draw the feasible set. Determine the vertices of the feasible set. The constraints can be read directly off the table above with 0.2x + 0.2y x + 0.2y x y 100 x 0, y 0 (I) A 00 y B C (III) These constraints can be rewritten as x + y 100 3x + y 3600 x + 2y 200 x 0, y 0 E 00 D (II) x The three boundary lines of the feasible set are x + y 100 3x + y 3600 x + 2y 200 (I) (II) (III) The vertices are A (0, 120) : B (00, 1000) : C (100, 40) : D (1200, 0) : E (0, 0) : y-intercept of line (III) intersection point of lines (I) and (III) intersection point of lines (I) and (II) x-intercept of line (I) the origin 2 (c) 7c Determine the number of number of bolts of each type that should be produced on the given in order to make the revenue as large as possible. Give the revenue in this case. Checking the value of the objective function R 1x + 20y at the vertices gives A (0, 120) : R 1 (0) + 20 (120) 2000 $20 B (00, 1000) : R 1 (00) + 20 (1000) 2700 $27 C (100, 40) : R 1 (100) + 20 (40) 2470 $247.0 D (1200, 0) : R 1 (1200) + 20 (0) $180 E (0, 0) : R 1 (0) + 20 (0) 0 Hence, the objective function is maximum at the point B (00, 1000) where the value is Thus the return is maximized by producing 00 lag bolts and 1000 carriage bolts. The maximum return is $27. Problem 9 continued...

11 Math 111 (071) Problems & s Page Problem 9 continued 3 (d) Rework this problem if it is required that at least as many lag bolts are produced as carriage bolts. Note that this requires an additional constraint which changes the feasible set. Thus the con- The additional constraint is that x y. straints can be written as x + y 100 3x + y 3600 x + 2y 200 x y 0 x 0, y 0 The resulting feasible set is shown. The boundary lines are (I) 00 y A B (IV) (III) x + y 100 3x + y 3600 x + 2y 200 x y 0 (I) (II) (III) (IV) D 00 C (II) x The vertices are A (70, 70) : B (100, 40) : C (1200, 0) : D (0, 0) : intersection point of lines (I) and (IV) intersection point of lines (I) and (III) x-intercept of line (II) the origin Note that line (III) does not bound the feasible set. The only vertex to test now is vertex A (70, 70) for which A (70, 70) : R 1 (70) + 20 (70) 2670 $267.0 This is larger than the values at the other vertices, as shown in part (c) above. Hence, with the additional constraint, the maximum return is obtained by producing 70 lag bolts and 70 carriage bolts for a maximum return of $267.0.

12 Problems & s Page Many mathematics educators invest in various securities, which is probably not a good idea these days. A recent survey of 10 mathematics educators revealed that 111 invested in stocks (T); 98 invested in bonds (B); 100 invested in GIC s (G); 80 invested in stocks and bonds; 83 invested in bonds and GIC s; 8 invested in stocks and GIC s; 9 did not invest in any of the three Define the three sets T, B, and G as the set of mathematics educators who invest in stocks, bonds, and GIC s, respectively, as indicated above. 3 (a) 8a Use the inclusion-exclusion principle, to find n (T G). n (A B) n (A) + n (B) n (A B) n (T) 111 n (B) 98 n (G) 100 n (T B) 80 n (B G) 83 n (T G) 8 n ( T B G ) 9 n (U) 10 Hence, n (T G) n (T) + n (G) n (T G) (b) 8b On the Venn diagram shown shade the region that represents the set T B G and explain what this set represents in terms of the investment portfolios of the mathematics educators surveyed. Use your result in part (a) to find n ( T B G ) T a f d c G b (1) g (80) e (3) B The set T B G represents those mathematicians who invest in bonds but not stocks or GIC s. Since n ( T B G ) n (T B G) 9 the number of mathematics educators who invest in at least one of stocks, bonds, or GIC s is n (T B G) n (U) n (T B G) But from the diagram it is clear that n ( T B G ) n (T B G) n (T G) Problem 10 continued...

13 Problems & s Page Problem 10 continued 2 (c) 8c Determine the number of mathematics educators who invest in stocks and bonds and GIC s. The set required is T B G. Labeling the basic regions in the three set Venn diagram in part (b) above in the usual way (See the diagram.) the result in part (b) gives b 1. Further, Hence, e 3. So that n (T B) 80 d + g and n (B) 98 b + d + e + g d + g + e 83 n (B G) 83 e + g g n (T B G) 80 3 (d) Consider the set consisting of the mathematics educators who invest in either stocks or bonds but not in GIC s. Represent this set using set theoretic notation and shade the region representing this set on the Venn diagram shown. Determine the number of mathematics educators in this set. T a (26) f () d (0) g (80) c b (1) e (3) B The set required is (T B) G. This set is shaded in the Venn diagram shown, and n (T B) G a + b + d From the results in parts (b) and (c) above b 1 and d 80 g 0. Further, n (T G) 8 f + g f G and Therefore, n (T) a + d + f + g a a 26 n (T B) G