Math 140 Exam II Review

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1 Setting Up Linear Programming Problems 1. Set up but do not solve the following linear programming problem. Math 140 Exam II Review Very Good Woodworking makes tables and desks. To produce each table requires 1 hour of labor, 10 square feet of wood, and 2 quarts of finish. To produce each desk requires 3 hours of labor, 20 square feet of wood, and 1 quart of finish. Available is at most 45 hours of labor, at most 350 square feet of wood, and at most 55 quarts of finish. The tables and desks yield profits of $4 and $3 each, respectively. Find the number of each product to be made in order to maximize profits. 2. Set up but do not solve the following linear programming problem. A College Station farmer has 10 acres available to plant maroon and orange carrots. Each acre of maroon carrots will yield 2 tons of carrots and each acre of orange carrots will yield 4 tons of carrots. He wants to have at least three times as many tons of maroon carrots as he does of orange carrots. The profit for maroon carrots is $200 per ton and hte profit for orange carrots is $100 per ton. How many acres of each type of carrots should the farmer plant to maximize his profit? 3. Set up but do not solve the following linear programming problem. An individual has at most $100,000 to invest in four funds: a money market fund, a bond fund, a conservative stock fund, and a speculative stock fund. These four investments are assumed to have annual returns of 6%, 8%, 10%, and 13%, respectively. The investor wants at least $10,000 in the conservative stock fund, at least $10,000 and at most $40,000 in the money market fund, the amount in the speculative stocks less than or equal to the amount in the money market, and finally the amount in conservative stocks less than or equal to the amount in bonds. Find the appropriate allocations of funds among these four investments to maximize the return. 1

2 Graphing Feasible Regions 4. What are the steps for graphing a feasible region? 5. Graph the feasible region of the following system: 3x + y 5 2x + 3y 8 6. Graph the feasible region of the following system: x 4y x y 23 2

3 7. How can you calculate the corner points of a feasible region? 8. Find the corner points for the system in problem #5. 9. Find the corner points for the system in problem #6. 3

4 Method of Corners 10. What is the Method of Corners used for, and what are the steps in the Method of Corners? x + y 5 x + y Minimize z = 4x + 5y subject to 2x + y 2 y 0 4

5 2x + y 20 x + y Minimize z = x + 3y subject to x + 3y Maximize z = 2x + 4y subject { x y 0 5

6 Simplex Tableau 14. Is the following a standard maximization problem? If so, set up the initial simplex tableau. 4x 3y 5 Maximize z = 4x 2y subject to the constraints 2x + y 9 x + 7y Is the following a standard maximization problem? If so, set up the initial simplex tableau. 4x 3y 5 2x + y 9 Maximize z = 4x 2y subject to the constraints x + 7y Is the following a standard maximization problem? If so, set up the initial simplex tableau. 4x 3y 5 2x + y 9 Maximize z = 4x 2y subject to the constraints x + 7y Find the next pivot for the simplex method for the following simplex tableau: x y z s 1 s 2 s 3 P

7 18. Find the next pivot for the simplex method for the following simplex tableau: x y z s 1 s 2 s 3 P Determine if the simplex tableau is in final form. If so, find the answer to the corresponding linear programming problem. If not, find the next pivot. x y z s 1 s 2 s 3 P Determine if the simplex tableau is in final form. If so, find the answer to the corresponding linear programming problem and determine if there were any leftover resources. x y z s 1 s 2 s 3 P Determine if the simplex tableau is in final form. If so, find the answer to the corresponding linear programming problem and determine if there were any leftover resources. x y z s 1 s 2 s 3 P

8 Sample Spaces & Events 22. An urn holds 10 identical balls except that 1 is white, 4 are black, and 5 are red. An experiment consists of selecting a ball from the urn and observing its color. What is a sample space for this experiment? Indicate the outcomes in the event "the ball is not white". 23. A restaurant offers six side dishes: rice, macaroni, potatoes, corn broccoli, and carrots. A customer must select two different side dishes for his dinner. What is the sample space? List the outcomes of the event "corn is selected". 24. A quality control engineer selects 8 transistors from the production line and notes how many are defective. What is the sample space? List the outcomes of the event "at least 6 are defective". 25. You have a collection of pencils and select one at random. Let E be the event that a pencil is 10 cm or longer and F is the event that the pencil is less than 25 cm. Describe the following sets: (a) E F (b) E C (c) E F C (d) (E F ) C 8

9 Probability 26. An experiment consists of selecting a letter from the word CONNECTICUT and observing it. What is the probability that "a vowel is selected"? 27. You draw a random card from a standard 52-card deck. What is the probability you drew any number between 3 and 5 inclusive? 28. An experiment consists of selecting a digit from the number and observing it. Create a probability distribution table for the experiment. 9

10 29. Let E and F be events with P (E) = 0.3, P (F ) = 0.5, and P (E F ) = 0.2. Find P (E C F ) and P (E C F C ). 30. At a computer plant, 2% of the monitors are defective, 3% of the keyboards are defective, and 1% of the computers have both defects. (a) Find the probability that a computer at this plant has at least one of these defects. (b) Find the probability that a computer at this plant has none of these defects. 10

11 Expected Value & Histograms 31. An insurance company sells a $24,000, three-year term life insurance policy to an individual for $1050. Find the expected return for the company if the probability that the individual will live for the next three years is A man wants to purchase a life insurance policy that will pay the beneficiary $30,000 in the event that the man s death occurs during the next year. The probability that the man will live another year is What is the minimum amount the man will pay for his premium? (Hint: The minimum premium amount occurs when the life insurance company s expected profit is zero) 33. Find the expected value of the random variable X, where the probability distribution of X is given by the following table: X P (X = x) Find the histogram for the random variable X, where the probability distribution of X is given by the following table: X P (X = x)