MAT183-2005Fall - MAT 183 Final Exam 14 December 2005...

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Unformatted text preview: MAT 183 Final Exam 14 December 2005 Signature: —__________ SU ID# Do all your work on this exam. State how you use your calculator. [Emblem l Possible points Score Subtotal u *5 F 1 20 —L 2 20 3 20 __*.__. 4 20 l— 5 20 l._ T— _T— a 6 20 l-—_ l 7 20 J l— l 8 20 + 9 20 1 10 20 L‘ *1— fl Total 200 Part I: Probability l. A survey of 80 college faculty who exercise regularly found that 35 jog, 25 swim, 10 cycle, 5 jog and swim, 4 swim and cycle, and 1 does all three. (a) (10 points) Let U denote the set of the 80 faculty surveyed and let C, J,S be the subsets of U consisting of the faculty who cycle, jog, and swim, respectively. Draw an appropriate Venn diagram and use the given data to determine the number of faculty in each of the eight basic regions. (b) (5 points) How many of the faculty members do not do any of these activities? (c) (5 points) How many just cycle? 0: 2. An urn contains 5 white balls and 4 red balls. A sample of 3 balls is selected (without replacement) from the urn, and the number of red balls in the sample is observed. (a) (10 points) What is the sample space for this experiment and what is the probability of each outcome? (b) (5 points) If W is the event “at least one white ball is selected” and R is the event “at least one red ball is selected,” compute Pr(W) and Pr(R). (c) (5 points) Are W and R independent? Why or why not? 3. A salesman sells cars with the probability .25. On a certain day he feels sick on arriving to work and decides that he will make at most 4 attempts to sell a car, and will go home as soon as he manages to sell a car, or if he fails to sell a car in all four attempts. (a) (10 points) Draw a tree diagram illustrating this experiment and put the probabilities along the branches. (b) (5 points) What is the probability that the salesman goes home without making a sale? (c) (5 points) Given that the salesman made a sale, what is the probability that he made at least 3 attempts to sell? CJ‘ 4. An archer who hits the bull’s eye of the target with the probability .3 receives $4 for hitting the bull’s eye and loses $1 for missing. Let X be the random variable whose values are the archer’s earnings in a series of 3 shots. (a) (10 points) Determine the probability distribution of X. (b) (5 points) Find the expected value of X. (c) (5 points) Find the standard deviation of X. Part II: Mathematics of Finance 5. A young couple buys their first home by borrowing $102,500. They arrange a 25-year mortgage with annual rate of interest 5.25% compounded monthly, and make monthly payments. (a) (10 points) What will their monthly payment be? (b) (10 points) How much will they still owe after faithfully making their payments for 8 years (96 months)? 6. At graduation (age 22), you decide to begin saving $50 each week for your retirement. (a) (10 points) Assuming that you deposit the money at 3.5% annual interest compounded weekly, how much will you have accumulated at age 62 (exactly 40 years later)? Assume that each year contains 52 weeks. (b) (10 points) Will the amount accumulated at age 62 be sufficient so that you could then withdraw $250 per week for the next 20 years? (Assume that the money on deposit will continue to earn 3.5% interest, compounded weekly.) Justify your answer. 7. If a savings account pays 4.25% annual interest compounded monthly, what will be the future value of each of the savings plans below? (a) (10 points) Deposit $10,000 and wait 15 years. (b) (10 points) Deposit $75 at the end of each month for 15 years. Part III: Linear Algebra 8. A street vendor sells handbags and watches. She has a stock of 350 total items. If she sells the handbags for $14 each and the watches for $10 each, how many of each did she sell if she sold all her stock for $4300? Solve this by completing the following steps. (a) (5 points) Set up the system of linear equations describing this situation. (b) (10 points) Solve your system of equations. (c) (5 points) Answer the problem in a complete sentence. 10 9. (a) (10 points) Use the method of Gaussian elimination to put the following system of equations into diagonal (reduced row—echelon) form. II I c: m+2y H o 2$+y (b) (10 points) Consider the following matrix, which has already had the method of Gaussian elimination applied to it. (Note that this is unrelated to part (30) Write down the general solution to the corresponding system of linear equations in 1:, y, and z. ll 10. In an economic system, two industries depend on each other for raw materials. To make $1 worth of processed lumber requires $.30 worth of lumber and $.10 worth of coal. To make $1 worth of coal requires $.10 worth of lumber and $.05 worth of coal. If worldwide demand for these products is for $1 billion worth of lumber and $2 billion worth of coal, what levels of production are required? Solve this problem by completing the following steps. (a) (5 points) Set up a matrix equation of the form (I — A)X = D to describe this input-output problem. (b) (5 points) Find (1 — A)‘1. (c) (10 points) Use your answer to solve for X. Calculator Commands and Formulas o STATS/EDIT/l: edit lists 0 STATS/CALC/l: lVarStats [list] gives basic statistics for data in list 0 MATH/PRB/4: 71!: n(n —1)(n — 2) - ~ - (2)(1) . MATH/PRB/2: nPr = (”1), . MATH/PRB/3: nCr = "(nil—)— 0 General solution of the difference equation yn = ayn_1 + b: 0 Future value (compound interest): F = (l + i)"P 0 Future value (annuity): . n _ 1 F = (1 + 1) R 2 0 Present value (annuity): _ (1 + i)" — 1 _ i(1 +1)" - MATRIX/NAMES: [A], [B], ...get matrix names for use in an expression 0 MATRIX/EDIT: create and edit matrices . MATRIX/MATH: — rowaap([matrix], rowA, rowB): interchange rowA and rowB in [matrix] — r0w+([matrix], rowA, rowB): add rowA t0 rowB in [matrix] - *row(c, [matrix], rowA): multiply rowA in [matrix] by the constant c — *r0w+(c, [matrix], rowA, rowB): add c times rowA to rowB in [matrix] — rref([matrix]): put [matrix] in reduced row-echelon form —— identity(N): the NXN identity matrix — det([n1atrix]): determinant of [matrix] 0 [A]‘1 the inverse of an invertible matrix 12 ...
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