Math 1300 SP2008 Final Exam: Solving Linear Equations, Probability and Statistics, Exams of Mathematics

Download Math 1300 SP2008 Final Exam: Solving Linear Equations, Probability and Statistics and more Exams Mathematics in PDF only on Docsity! Math 1300 Spring 2008 Final Exam May 15, 2008 3:30-5:30pm No work = No credit Name:_______________________________ Student Number:_____________________ Section Number:__________________ Instructor:_______________________________ Signature:________________________________________________ Problem Points Possible Student’s Score 1 10 2 8 3 18 4 25 5 32 6 18 7 24 8 24 9 16 10 25 Total 200 Instructions: You have one hour to complete this exam. You may use a calculator, but you may not use a graphing calculator. You may not use any notes, books, or other media or any purpose during this exam. Any act of academic dishonesty which is in any way related to this exam will result in a score of zero on this exam for each student involved. Furthermore, a report of the incident will be made to the appropriate university authorities for further disciplinary action. Potentially Helpful Formulas: ( ) ( ) ( ) ( ) ( )2 1 (1 ) 1 1 1 1 1 1 1 1 n i n i i m n n n i i i I A P rt A P i APY i i FV PMT PV PMT i i x f x s n Prt x x f n = − = = + = + = + + − − + = − − = = = ∑ ∑ i − show all work…show all work…show all work…show all work…show all work…show Page 1 of 10 Math 1300 SP2008 Final Exam May 15, 2008 3:30-5:30 No work = No credit Name:_______________________________ Student Number:_____________________ Problem 1: (10 points): A “payday loan” is a loan designed to provide short-term financial assistance to individuals who are short of funds between paydays. A company in Columbia, Quik Cash, offers payday loans for up to 14 days at the cost of $9.00 for every $500 borrowed (regardless of the length of the loan). Suppose someone borrows $1,500 for 8 days from this company. What is the annual simple rate of interest for this loan? 827 1,500 360 0.81 PrtI r r ⎛ ⎞= ⎜ ⎟ ⎝ ⎠ = = Problem 2: (8 points) A “title loan” is a type of short-term consumer loan in which the borrower signs over his or her vehicle as collateral. For title loans, suppose that Quik Cash charges 25% interest per month. Find the APY for a title loan, given that the interest rate per month is 25%. 12 APY (1 ) 1 APY (1.25) 1 13.55 1,355% mi= + − = − = all work…show all work…show all work…show all work…show all work…show Page 2 of 10 Math 1300 SP2008 Final Exam May 15, 2008 3:30-5:30 No work = No credit Name:_______________________________ Student Number:_____________________ Problem 5: (32 points) An electronics firm manufactures two types of personal computers, a desktop model and a laptop model. The production of the desktop requires a capital expenditure of $400 and 40 hours of labor. The production of the laptop requires a capital expenditure of $250 and 30 hours of labor. The firm has $20,000 capital and 2,160 labor-hours available for production of the desktop and laptop computers. Finally, each desktop computer contributes a profit of $320, and each laptop model contributes a profit of $220. Let number of desktop computers produced number of laptop computers producedy x = = (a) (4 points) Let P represent the total profit to the company from producing x desktop computers and y laptop computers. Find a formula for P. 320 220P x y= + (b) (8 points) Write down all linear constraints which x and y must satisfy given the information in the problem. 2160 400 250 40 30 20,000 , 0 x y x y x y ≤ + ≤ ≥ + (c) (14 points) Graph the feasible set for the system of inequalities from part (b). Label the feasible set by writing “FS” inside of the feasible set. Find the coordinates of all corner points and label all corner points with the appropriate coordinates. all work…show all work…show all work…show all work…show all work…show Page 5 of 10 Math 1300 SP2008 Final Exam May 15, 2008 3:30-5:30 No work = No credit Name:_______________________________ Student Number:_____________________ (d) (6 points) Construct a corner point table which lists each of the corner points and the corresponding values for the objective function. How many desktop computers and how many laptop computers should the company produce to maximize profit? Corner Point Profit (0,72) $15,840 (30,32) $16,640 (50,0) $16,000 The company should produce 30 desktop and 32 laptop computers to maximize profit. Problem 6: (18 points) Suppose that the matrix equation 4 6 2 1 3 0 2 2 1 2 2 1 C −⎡ ⎤ ⎡ = ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ is true. (That is, the matrix product on the left equals the matrix on the right.) (a) (4 points) What is the size of the matrix C? 2x4 (b) (8 points) Find the inverse of the matrix 4 6 2 2 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ . 2 2 1 1 2 1 1 1 2 2 1 2 1 1 42 2 1 4 6 1 0 4 6 1 0 0 2 1 2 0 11/ 2 1 2 2 0 1 1 1 0 1/ 2 1 1 0 1/ 2 1 1 0 1/ 2 0 1 1/ 2 1 1 0 1/ 2 3 / 2 1 0 1/ 2 3 / 2 0 1 1/ 2 1 R R R RR R R R R R R R → →− + → − + → ↔ ⎡ ⎤ ⎡ ⎤ ⎡ − ⎤ ⎡ ⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯→⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎡ − ⎤ ⎡ − ⎤ ⎯⎯⎯⎯⎯→ ⎯⎯⎯→⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ − ⎤ ⎥ ⎦ ⎤ ⎥− − ⎦ (c) (6 points) Find the matrix C. 1/ 2 3 / 2 2 1 3 0 1/ 2 7 / 2 3 / 2 3 / 2 1/ 2 1 1 2 2 1 0 5 / 2 1/ 2 1 C − −⎡ ⎤ ⎡ ⎤ ⎡ = =⎢ ⎥ ⎢ ⎥ ⎢− −⎣ ⎦ ⎣ ⎦ ⎣ all work…show all work…show all work…show all work…show all work…show Page 6 of 10 Math 1300 SP2008 Final Exam May 15, 2008 3:30-5:30 No work = No credit Name:_______________________________ Student Number:_____________________ Problem 7: (24 points) A banquet will consist of 4 meat dishes and 3 vegetarian dishes, to be served one dish at a time. The order that the dishes will be served will be selected at random, so the dishes could come in any order. An outcome is a list of the dishes in the order in which they are to be served. Let all meat dishes are served consecutively and all vegetarian dishes are served consecutively. a meat dish is served first E F = = (a) (4 points) Find the number of outcomes in the sample space. (7,7) 7! 5040P = = (b) (6 points) Find Pr( )E . ( ) 2(4!)(3!) 288Pr( ) 0.0571 ( ) 5040 5040 n EE n S = = = = (c) (6 points) Find Pr( )F ( ) 4(6!) 2880Pr( ) 0.571 ( ) 5040 5040 n FF n S = = = = (d) (8 points) Find Pr( | ).E F Are the events E and F independent? Justify your answer. Pr( ( (4!)(3!) 144 1Pr( | ) 0.05 Pr( ) ( ) 4(6! ) ) 80 2 ) 28 0 FE n EE F F n F F = = = = = = ∩ ∩ No, because Pr( | ) Pr( )E F E≠ all work…show all work…show all work…show all work…show all work…show Page 7 of 10