Math 1070 Final Exam Practice Spring 2014

Transcription

1 University of Connecticut Department of Mathematics Math 1070 Practice Spring 2014 Name: Instructor Name: Section: Read This First! This is a closed notes, closed book exam. You can not receive aid on this exam from anyone else - No sharing of calculators! If you have a question, ask your instructor. Make sure your exam has 9 question pages. If part of the solution is written outside the space provided, then clearly indicate this. If you finish this exam early, check all your answers before handing in your copy. Once a copy is handed in, it cannot be asked back. Some partial credit may be given depending on the correctness of the work submitted. You must show all work and calculations needed to reach your answers. Just using a calculator is not sufficient for credit. Grading - For Administrative Use Only Page: Total Points: Score:

2 1. The Easter bunny will supply 18 Easter eggs at a price of $1.80 each. If the bunny is offered [6] $1.50 for each egg, then they will supply 4 fewer eggs. The demand for Easter eggs is given by p = 0.75x + 6. Find the supply equation, and the equilibrium point. 2. A baker has 15 pounds of sugar to make doughnuts and fritters. Each donut requires.07 pounds of sugar, while each fritter requires.125 pounds of sugar. The baker wants to have at least three times as many doughnuts as fritters. Doughnuts will sell for $0.75 each, while fritters sell for $1.50. How many doughnuts and fritters should be made to maximize revenue? (a) Write down the inequalities describing this scenario. [4] Page 1 of 9

3 (b) Graph the feasible region and find its corner points. [6] (c) Determine how many doughnuts and fritters should be made to maximize revenue. [4] Page 2 of 9

4 3. A survey of business executives found that 40% read Business Week, 50% read Fortune, 40% read Money, 17% read both Business Week and Fortune, 15% read both Business Week and Money, 14% read both Fortune and Money, and 8% read all three of these magazines. (a) Draw a Venn diagram representing this information. [4] (b) What is the probability that one of these executives reads exactly one of these three [2] magazines? (c) What is the probability that one of these executives reads at least one of these three [3] magazines? (d) What is the probability that one of these executives reads none of these three magazines? [3] 4. We roll a fair, six-sided die three times. What is the probability of getting a 4 on the first roll, [3] an odd number on the second roll, and a number greater than or equal to 3 on the third roll? Page 3 of 9

5 5. Let E and F be two events with P (E) = 0.4, P (F ) = 0.5, and P (E F ) = 0.7. (a) Draw a Venn diagram of all the probabilities. [3] (b) Find P (E c F ). [2] (c) Find P (F E). [3] 6. A special class of license plate is being made, consisting of 4 digits (with no repitition) followed [4] by 2 letters, one a vowel and the other a consonant. (A vowel is from the set {a, e, i, o, u}, and a consonant is any non-vowel.) How many possible license plates are there? 7. In a group of 12 people, 4 are traveling to NH, 4 to MA, 3 are staying in CT, and 1 is going to [4] NY. In how many ways can this be done? 8. Mario is searching for coins in various boxes, each of which may or may not contain one coin. [4] Suppose that the probability of a box containing a coin is 60%. If he opens 10 boxes, what is the probability of getting at least 8 coins? Page 4 of 9

6 9. A firm has four machines that produce the same component. Machine A, B, C, and D supply 20%, 30%, 40%, and 10% respectively, of the components made. However sometimes the parts are defective. Specifically, 1% of the parts produced by machine A are defective, 2% of the parts from machine B, 3% from machine C, and 4% of the parts from machine D are defective. Suppose an experiment consists of choosing a machine, then selecting a random component made from that machine. (a) Draw a tree diagram describing the sample space in this problem, showing all probabilities. [4] (b) What is the probability that a randomly selected component will be defective? Show the [3] formula you use and all calculations. (c) Given that a component is defective, what is the probability it came from machine D? [3] Page 5 of 9

7 10. A box contains six red and seven white balls. Three balls will be chosen, without replacement. If [6] X represents the number of red balls chosen, find the expectation E(X) and sketch a histogram for X. 11. Is it more profitable for a person to receive $2,000 now or $2,700 in 6 years? Assume that they [4] can earn 6% interest compounded monthly on the money they receive now. 12. Jim s dad wants to send Jim to college. Jim is 2 years old today. Assume Jim will go to college [6] in 16 years. Jim s dad wants $100,000 for Jim s education. How much should Jim s dad save every month, if the bank offers 5% interest compounded monthly? Page 6 of 9

8 13. The following augmented matrix [4] comes up in your calculation while trying to solve a system of linear equations. Determine whether this system has one unique solution, infinitely many solutions, or no solutions. 14. Consider the system of equations below. 2x + y + z = 1 3x + 2y + z = 2 2x + y + 2z = 1 (a) Write the system in the form AX = B for matrices A, X, and B. [3] (b) Solve the above system using the inverse of a matrix. Find the inverse of the appropriate [12] matrix by using the Gauss-Jordan method. Show all row operations that you use. Page 7 of 9

9 Formulas From Chapter F Simple Interest F = P (1 + rt) r eff = r 1 rt Compound Interest ( F = P 1 + m) r mt = P (1 + i) n ( r eff = 1 + m) r m 1 = (1 + i) m 1 Future Value of Annuities FV = PMT (1 + i)n 1 i Present Value of Annuities PV = PMT 1 (1 + i) n i Page 8 of 9

10 Page 9 of 9