Final Exam Review for Week in Review. a) Consumers will buy units of a certain product if the price is $5 per unit. For each decrease of $3 in the price, they will buy more units. Suppliers will provide none unless the price per unit is $5 or more. At a price of $2 per unit, they will provide 25 units. a) Find the demand price, p, as a function of the quantity, x. b) Find the equilibrium point. c) For a different product, demand and supply equations are 3p.2x 65 and 2p. x 6. Find the equilibrium point.
2. A company produces two products, A and B. Each unit of A costs $5 to produce and uses 3 cu. ft. of storage space. Each unit of B costs $7 to produce and uses 5 cu. ft. of storage space. Write two matrix products that each show the total cost and total storage space used if they produce 8 units of A and 6 units of B. Do not compute the product. 2 4 3. A 3 5 2 T a) Find 2 A B. 2 5 6 B x y b) Find AB 4. 3 3 4 5 6 3 2 4 a 2 3 5 5 8 3 b 6 7 C D 3 2 x 2 4 2 5 4 6 3 y 7 2 5 6 3 2 Find the entry in row 3 column 2 of CD.
5. 2 3 4 2 5 3 8 6 4 2 A a) Perform the first pivot in the row reduction of matrix A. Indicate your row operations clearly. b) What is wrong with the following method to pivot A on row column? R->R3+R2, R2+3R, R4-4R 6. Each augmented matrix represents a system of equations in x, y, and z. Give the complete solution set and 2 particular solutions or state no solution. a) 3 4 2 5 b) 3
7. A company produces two products, X and Y. Each uses time in the assembly, finishing and packaging departments. The requirements in minutes are shown in the table. The company wants to use at most 2 minutes in assembly, 85 minutes in finishing and 6 minutes in packaging for these two products. Profits are $6 per unit of X and $4 per unit of Y. How many of each of X and Y should they produce to maximize profits and stay within their time limits? X Y time in minutes Assembly 8 4 Finishing 5 5 Packaging 3 4 8. S is the region described by 3x 2y 9, 2x 3y 8, x, y. R 3x 4y. Find the min and max of R on S or state DNE.
9. a) A company advertises its product through television, radio and magazines. people who use the product were asked if they learned about it through these means. 5 had learned about it through none of these. 3 learned of it through TV only. 49 learned of it through TV or radio, possibly both, but did not read the magazine ads. 3 had heard both the TV and radio ads. 9 had been exposed to all three types of ads. had only read the magazine ads. 2 had read the magazine ads and seen the TV ads. Each letter in the Venn diagram represents the number of elements in its region. T is the set of people who saw the TV ads, R is the set of people who heard the radio ads and M is the set of people who read the magazine ads. Find the value of each lower case letter in the Venn diagram and write it in the diagram. b) people were asked if they saw the TV ad. Later they were asked if they heard the radio ad. 8 said they had seen the TV ad, 7 had heard the radio ad and 5 had heard neither. How many heard both ads? How many saw the TV ad but did not hear the radio ad?
. An experiment consists of drawing 5 cards from a box containing 36 cards. Each card has one symbol, a letter of the alphabet or a digit. All the 26 letters and all ten digits are represented. I) Assume the cards are chosen without replacement and kept in order. a) How many outcomes are in the sample space? b) How many outcomes correspond to the event that exactly two letters and exactly 3 digits are chosen? II) Assume the cards are chosen all at once and without replacement. No order is observed. a) How many outcomes are in the sample space? b) How many outcomes correspond to the event that at least 3 digits are chosen? III) Assume the cards are chosen with replacement and kept in order a) How many outcomes are in the sample space? b) How many outcomes correspond to the event that at least one letter is chosen?
. An 8-long code is made from the digits and the 26 letters. How many codes are there if: a) there are exactly 5 digits and 3 letters if the letters may repeat but the digits may not repeat? b) the code has only letters and exactly 3 of the spaces are a s and there are no other repeats? c) the code has only digits and at least two of the spaces have the same digit? 2. A person chooses 5 cards from a standard 52-card deck. Find the probability he gets: a) exactly 2 face cards or exactly 2 aces. (still choosing 5) b) at least 4 hearts. c) at least one spade.
3. A trial experiment is done to test the effectiveness of treating a medical condition with a vitamin or a medicine. 9 people are divided into groups of 3 each. One group of people take the vitamin, (group V), one group take the medicine, (group M) and one third are given a placebo, (group P). 8 in the vitamin group were cured, 24 in the medicine group were cured, and 2 in the placebo group were cured. Let C be the event that a person is cured. Express all answers as exact decimals or as fractions. Do not round or truncate. a) Make a tree diagram or a table for this information. b) According to this experiment and your tree or table, which of the events V, M and P is independent of being cured? c) What is the probability someone in the group took the placebo and was cured? d) What is the probability someone who was cured, took the medicine.
4. Three security checkpoints are independent of each other. The probabilities that an unauthorized person will get by the checkpoints are.3,.5, and.4 respectively. Find the probability that a person who is unauthorized a) will pass through checkpoint 3 but not checkpoints or 2. b) will pass through all three checkpoints. c) will be stopped by at least one checkpoint. 5. A bag contains 4 nickels and 3 quarters. A person chooses 2 coins at the same time. The random variable, X, is the number of quarters chosen. a) Write the probability distribution of X. b) If he keeps all the coins he selects, find his expected winnings.
6. A certain mutation of a gene occurs in 5% of a given population. 7 people are randomly selected. X is the number among the 7 who have the mutation. a) Find P(8 < X < 2) rounded to 4 decimal places. b) Find the expected value and variance of X, and the standard deviation of X. 7. The random variable, X, is normally distributed with mean 5 and unknown standard deviation. It is known that P(5<X<65) =.29. Find P(X<35). 8. Lengths of babies in a certain city are normally distributed with mean 2 inches and standard deviation 2 inches. a) What portion of babies are between 8 and 24 inches? b) Find a length, L, so that 6% of babies are longer than L. c) Find a length, l, so that 5% are shorter than l.
9. To purchase a $3, car, a down payment of $6 is made. The rest is paid in monthly installments over 5 years. Annual interest is 6.5% and interest is compounded monthly. a) Find the monthly payment. b) How much interest will be paid over the whole 5 years? c) How much is still owed at the end of 2 years? d) How much of the 25 th payment is interest? 2. A person wants to be able to withdraw $ per month for 5 years beginning 3 years from now. He assumes he can get an annual interest rate of 4% compounded monthly for the whole 45 years. a) How much will he need to have in the account 3 years from now (At the beginning of the 5 years of monthly withdrawals)? b) How much should he deposit today to have this amount in 3 years? c) If instead he wants to make monthly deposits to accumulate the required amount in 3 years, how much should these deposits be? Assume no initial deposit.