MAT 112 Final Exam Review

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1 MAT 2 Final Exam Review. Write the slope-intercept form of the equation of the line that passes through the points ( 2, 9) and (6, 7). Then find the x-intercept, the y-intercept, and give the y-coordinate of the point on the line that has xcoordinate A donut shop sells glazed donuts for $0.76 each. The daily fixed costs are $90, and the variable costs are $0.2 per donut. Let x represent the number of donuts sold each day. Find the cost equation C, the revenue equation R, and the number of donuts that must be sold each day for the donut shop to break even. 3. A company sells exercise DVDs for $46.20 each. The monthly fixed costs are $43, 60, and the variable costs are $6.60 per DVD. Let x represent the number of DVDs sold each month. Find the cost equation C, the revenue equation R, and the number of DVDs that must be sold each month for the company to break even. 4. Mark buys a sport jet ski for $3, 000 and assumes it will have a trade in value of $32, 330 after 3 years. Find a linear model (V = mt + b) for the depreciated value V of the jet ski t years after it was purchased. Then give the depreciated value of the jet ski after years, and determine when the depreciated value of the jet ski will be $22, Find the vertex and x-intercepts of the quadratic function f(x) = x 2 +7x+. Approximate the coordinates of these points accurate to two decimal places. 6. The manufacturer of a smart phone determines that their revenue for selling x thousand phones is R(x) = 240x 3.7x 2 thousand dollars. How many phones must be sold to maximize revenue, and what is the maximum revenue? 7. The manufacturer of a popular video game determines that the total cost for producing x thousand games is C(x) = 7x thousand dollars, and their revenue for selling x thousand games is R(x) = 40x.7x 2 thousand dollars. Find and simplify the profit function P (x), and determine how much profit results from producing and selling 44 thousand games. 8. The manufacturer of a widescreen LED television determines that the total cost for producing x thousand televisions is C(x) = x + 33 thousand dollars, and their revenue for selling x thousand televisions is R(x) = 297x 2.7x 2 thousand dollars. Find and simplify the profit function P (x), and determine which values of x give a profit. 9. Find the interest rate on a loan of $4, 000 that charges $6, 2 of simple interest after years. 0. How much should be invested now at 8.7% simple interest if $2, 80 is needed in 2 years?. How long will it take for $9, 000 to grow to $2, 37 at 6.2% simple interest?

2 2. At the birth of their child, John and Cindy Newparents deposit $0, 000 into an account to save for their daughter s future wedding expenses. What will be the balance of this account in 20 years if it yields 6.2% interest compounded semiannually? 3. If an investment earns 8.% compounded monthly, how much should you deposit now to have $7, 00 in 27 years? 4. A married couple invests $27, 000 to save for the down payment on a new house. How long will the money need to be invested at 4% compounded quarterly to grow to $3, 000?. Find the annual percentage yield (APY) for the following accounts. Record enough decimal places of accuracy to show the difference between the two accounts. Account : An account that yields 4.7% interest compounded semiannually. Account 2: An account that yields 4.7% interest compounded monthly. Which account is the better investment? 6. Guaranteed Income Life offered an annuity that pays 6.7% compounded semiannually. If $600 is deposited into this annuity every six months, how much is in the account after 6 years? How much of this is interest? 7. Parents have set up a sinking fund in order to have $, 000 in years for their children s college education. How much should be paid every six months into an account paying 7.7% compounded semiannually? What is the amount of interest that would be earned over the year period? 8. A family has a $82, 000, 20-year mortgage at.2% interest compounded monthly. What monthly payment will amortize the mortgage in 20 years? What is the unpaid balance on the house after 0 years? What is the unpaid balance after years? 9. An investment firm offers a 2-year ordinary annuity with a guaranteed rate of 3.87% compounded monthly. How much should you pay for one of these annuities if you want to receive payments of $4, 900 monthly over the 2-year period? 20. Solve the following system of linear equations using an algebraic method. You must show all steps of the algebraic process used. x + 6y = 8x 6y = Solve the following systems of linear equations. If there are an infinite number of solutions, then give a parameterization of the solution set. x + 9y = 22 9x + y = 74 9x + y = 9 4x 6y = 49 x 4y = 7 2x + 8y = 4

3 22. A package delivery service charges a base price for delivery of packages weighing pound or less, and a surcharge for each additional pound. A customer is billed $3 for shipping a 7 pound package and $9 for shipping a 27 pound package. Find the base price and the surcharge for each additional pound. 23. Animals in an experiment are to be kept under a strict diet. Each animal should receive 7 grams of protein and 4 grams of fat. The laboratory technician is able to purchase two food mixes: Mix A has 30% protein and 20% fat; mix B has 26% protein and 4% fat. How many grams of each mix should be used to obtain the right diet for one animal? 24. Perform the row operation R R 2 on the following matrix. [ ] Perform the row operation R R on the following matrix. [ ] Perform the row operation 4R 2 + R R on the following matrix. [ ] Identify the row operations performed below. [ ] [ ] [ ] [ ] [ ] 3 a 3 27 b 3 27 c 3 27 d Graph the solution regions to the systems of linear inequalities and list the corner points of the solutions. 3x + 3y 8 x 4y 2 2x + 7y 33 3x + 8y 72 3x + 2y 36 x 0 y Find the maximum and minimum value of P = 3x + 9y subject to x + y 2 4x + y 60 x 0 y 0 Also, give the x and y coordinates of the points in the region at which these maximum and minimum values occur.

4 28. A rancher raises goats and llamas on his 220-acre ranch. Each goat needs 2 acres of land and requires $00 of veterinary care per year, and each llama needs 2 acres of land and requires $300 of veterinary care per year. The rancher can afford no more than $7, 400 for veterinary care this year. If the expected profit is $20 for each goat and $80 for each llama, then how many of each animal should the rancher raise to obtain the greatest possible profit? What is the maximum profit? 29. A farmer grows wheat and barley on her 20-acre farm. Each acre of wheat requires 4 days of labor per year, and each acre of barley requires 2 days of labor per year. The farmer can provide no more than 320 days of labor this year. If the expected profit is $40 for each acre of wheat and $30 for each acre of barley, then how many acres of each should the farmer grow to obtain the greatest possible profit? What is the maximum profit? 30. Refer to the Venn diagram below. Find n(a), n(a ), n(a B), n((a B) ), and n(a B). U A B Among a group of 74 high school athletes, 33 play soccer, 39 run cross-country, and 8 participate in both soccer and cross-country. How many of these students participate in soccer or cross-country? How many of these students participate in neither soccer or cross-country? 32. You would like to make a salad that consists of lettuce, tomato, cucumber, and mushrooms. You go to the supermarket intending to purchase one variety of each of these ingredients. You discover that there are six varieties of lettuce, four varieties of tomatoes, six varieties of cucumbers, and five varieties of mushrooms for sale at the supermarket. How many different salads can you make? 33. How many 6-letter code words are possible from the first letters of the alphabet if (d) no letters are repeated? letters can be repeated? adjacent letters must be different? no letters are repeated, the first letter must be a consonant, and the last letter must be a vowel.

5 34. How many five-card hands from a standard 2-card deck (d) will contain only face cards? will not contain any face cards? will contain only kings and aces? will not contain any kings? 3. A committee consisting of a president, vice president, and secretary is to be chosen from a class of 24 students. How many such committees are there? 36. How many ways can a 4-person committee be selected from a group of 20 people? 37. A card is drawn at random from a standard 2-card deck. Events G and H are G = the card drawn is red. Find the probability of H given G. H = the card drawn is odd. (face cards and aces are not valued). 38. An eleven player offensive unit for a high school football team is to be chosen from a roster of 23 players. What is the probability of any particular choice? 39. A center, a guard, and a forward are to be chosen from a basketball team with 0 players. What is the probability of any particular choice? 40. You will take either a basket weaving class or a philosophy class, depending on what your advisor decides. You estimate that the probability of getting an A in basket weaving is 0.24 (it is a hard class), while in philosophy your probability of getting an A is However, your major is straw construction, and the chance of your advisor choosing the philosophy class is only 38%. What is the probability of that you get an A? 4. A box contains 30 blue marbles and 90 red marbles. You reach in and pull out a marble, note its color and put it back. Then you pull out a second marble. What is the probability that the first marble is blue and the second is red? 42. A two card hand is dealt from a standard 2-card deck. Then the deck is shuffled and another two card hand is dealt. What is the probability that both of the hands contain no aces? 43. Find the mode(s), mean, median, range, and standard deviation accurate to three decimal places, and construct a histogram using class intervals of width starting at 70. for the following data set consisting of 6 data values

6 44. Thirteen thousand tickets are sold at $ each for a charity raffle, and there is one first prize worth $8, 000, five second prizes worth $800, and 40 third prizes worth $80. Let X be the random variable for the amount won on a single raffle ticket. Construct the probability distribution of X, and find the expected value of a ticket accurate to two decimal places. 4. A pair of dice are rolled and you win $7 if the sum of the faces is less than or equal to four, you lose $ if the sum is between four and eight, and you win $3 if the sum is greater than or equal to eight. Let X be the random variable for the amount won on a single play of this game. Construct the probability distribution of X, and find the expected value of the game accurate to two decimal places. (, ) (, 2) (, 3) (, 4) (, ) (, 6) (2, ) (2, 2) (2, 3) (2, 4) (2, ) (2, 6) (3, ) (3, 2) (3, 3) (3, 4) (3, ) (3, 6) (4, ) (4, 2) (4, 3) (4, 4) (4, ) (4, 6) (, ) (, 2) (, 3) (, 4) (, ) (, 6) (6, ) (6, 2) (6, 3) (6, 4) (6, ) (6, 6) 46. If the probability of a new employee in a fast-food chain still being with the company at the end of year is 0.3, what is the probability that out of 28 newly hired people exactly 3 will be with the company after year? What is the probability 9 or more will still be with the company after year? Give these probabilities accurate to four decimal places. 47. A multiple-choice test is given with 4 choices (only one is correct) for each of 2 questions. What is the expected number of correct answers that would result from randomly guessing? What is the probability of getting at least correct by randomly guessing on all questions? Give this probability accurate to four decimal places.

7 Answers to MAT 2 Final Exam Review. Slope-intercept ( form ) of the equation of the line: y = 2x x-intercept:, 0 y-intercept: (0, ) y-coordinate of the point with x-coordinate 22: C = 0.2x + 90 and R = 0.76x. 37 donuts must be sold each day to break even. 3. C = 6.6x and R = 46.2x. 4, 80 DVDs must be sold each month to break even. 4. The linear model is V = 90t After years the value of the jet ski will be $4, 00. The value of the jet ski will be $22, 790 after 9 years.. Vertex: (.7, 9.4) x-intercepts: ( 0.33, 0) and ( 3.07, 0) thousand smart phones must be sold to maximize revenue. The maximum revenue is $3, 840, P (x) =.7x x 827. The profit is $637, 000 when 44 thousand games are produced and sold. 8. P (x) = 2.7x x 33. There is a profit when 6 x % 0. $22, years 2. $34, $, years (or 3 years and 2 quarters). APY of Account : % APY of Account 2: % Account is better for an investment. 6. Future Value: $8, Interest Earned: $ $ Amortization Payments: $3, Unpaid balance after 0 years: $36, Unpaid balance after years: $206, $940, (, 9) 2. (8, 2) No solutions ( Infinite number of solutions parameterized by (4t 7, t) or t, t + 7 ) Base price: $7 Surcharge: $ grams of Mix A and 20 grams of Mix B

8 24. [ ] [ ] [ ] a. R R b. R 2 + R R 2 c. R 2 R 2 d. R 3R 2 R 26. y y Corner points: (8, 7), and (4, 2). x x Corner points: (0, 9), (0, 8), and (8, 6). 27. Maximum Value: 008 when x = 0 and y = 008 Minimum Value: 20 when x = 40 and y = goats and 8 llamas Maximum Profit: $4, acres of wheat and 80 acres of barley Maximum Profit: $4, n(a) = 73, n(a ) = 4, n(a B) = 97, n((a B) ) = 30, and n(a B) = participate in soccer or cross-country 20 participate in neither , 640, 77, 6, 00, 000 (d) 72, , (d), 72, , , 84

9 Mode: 78 Mean: 83.2 Median: 82 Range: 8 Standard Deviation: Histogram: Probability Distribution: x i p i 3000 E(X) = $ Probability Distribution: x i 7 3 p i 6 2 E(X) = $ Probability exactly 3 are still with the company: Probability 9 or more are still with the company: Expected number of correct answers: 2 4 = 6.2 Probability of getting at least correct: