Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)
14E15E16E17ESolving a System of Equations in Three Variables Find the complete solution of the linear system, or show that it is inconsistent. 18. {xy+z=0y+2z=2x+yz=2Solving a System of Equations in Three Variables Find the complete solution of the linear system, or show that it is inconsistent. 19. {x+2yz=6y3z=16x3y+2z=1420E21E22ESolving a System of Equations in Three Variables Find the complete solution of the linear system, or show that it is inconsistent. 23. {x4z=12xy6z=42x+3y2z=8Solving a System of Equations in Three Variables Find the complete solution of the linear system, or show that it is inconsistent. 24. {xy+2z=23x+y+5z=82xy2z=725E26ESolving a System of Equations in Three Variables Find the complete solution of the linear system, or show that it is inconsistent. 27. {2y+4z=12x+y+2z=14x2y=028E29E30E31E32ESolving a System of Equations in Three Variables Find the complete solution of the linear system, or show that it is inconsistent. 33. {x+yz=0x+2y3z=32x+3y4z=334E35ESolving a System of Equations in Three Variables Find the complete solution of the linear system, or show that it is inconsistent. 36. {2x+4yz=3x+2y+4z=6x+2y2z=037E38EFinancial Planning Mark has 100,000 to invest. His financial consultant advises him to diversify his investment in three types of bonds: short-term, intermediate-term, and long-term. The short-term bonds pay 4%, the intermediate-term bonds pay 5%, and the long-term bonds pay 6% simple interest per year. Mark wishes to realize a total annual income of 5.1%, with equal amounts invested in short- and intermediate-term bonds. How much should he invest in each type of bond?Financial Planning Cyndee wants to invest 50,000. Her financial planner advises her to invest in three types of accounts: one paying 3%, one paying 512, and one paying 9% simple interest per year. Cyndee wants to put twice as much in the lowest-yielding, least-risky account as in the highest-yielding account. How much should she invest in each account to achieve a total annual return of 2540?Agriculture A farmer has 1200 acres of land on which he grows corn, wheat, and soybeans. It costs S45 per acre to grow corn, S60 to grow wheat, and S50 to grow soybeans. Because of market demand, the farmer will grow twice as many acres of wheat as of corn. He has allocated S63,750 for the cost of growing his crops. How many acres of each crop should he plant?Gas Station A gas station sells three types of gas: Regular for 3.00 a gallon, Performance Plus for 3.20 a gallon, and Premium for 3.30 a gallon. On a particular day 6500 gallons of gas were sold for a total of 20,050. Three times as many gallons of Regular as Premium gas were sold. How many gallons of each type of gas were sold that day?Nutrition A biologist is performing an experiment on the effects of various combinations of vitamins. She wishes to feed each of her laboratory rabbits a diet that contains exactly 9 mg of niacin, 14 mg of thiamin, and 32 mg of riboflavin. She has available three different types of commercial rabbit pellets; their vitamin content (per ounce) is given in the table. How many ounces of each type of food should each rabbit be given daily to satisfy the experiment requirements?Diet Program Nicole started a new diet that requires each meal to have 460 calories, 6 g of fiber, and 11 g of fat. The table shows the fiber, fat, and calorie content of one serving of each of three breakfast foods. How many servings of each food should Nicole eat to follow her diet?Juice Blends The Juice Company offers three kinds of smoothies: Midnight Mango, Tropical Torrent, and Pineapple Power. Each smoothie contains the amounts of juices shown in the table. On a particular day the Juice Company used 820 oz of mango juice, 690 oz of pineapple juice, and 450 oz of orange juice. How many smoothies of each kind were sold that day?Appliance Manufacturing Kitchen Korner produces refrigerators, dishwashers, and stoves at three different factories. The table gives the number of each product produced at each factory per day. Kitchen Korner receives an order for 110 refrigerators, 150 dishwashers, and 114 ovens. How many days should each plant be scheduled to fill this order?Stock Portfolio An investor owns three stocks: A, B, and C. The closing prices of the stocks on three successive trading days are given in the table. Despite the volatility in the stock prices, the total value of the investors stocks remained unchanged at 74,000 at the end of each of these three days. How many shares of each stock does the investor own?Electricity By using Kirchhoffs Laws, it can be shown that the currents I1, I2, and I3 that pass through the three branches of the circuit in the figure satisfy the given linear system. Solve the system to find I1, I2, and I3. {I1+I2I3=016I18I2=48I2+4I3=549EIf a system of linear equations has infinitely many solutions, then the system is called _____. If a system of linear equations has no solution, then the system is called _____.2E3E4E5E6E7E8E9E10E11E12E13E14E15EForm of a Matrix A matrix is given, (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. 16. [107001300001]17E18E19E20EElementary Row Operations Perform the indicated elementary row operation. 21. [112031141211] Add 3 times Row 1 to Row 2.Elementary Row Operations Perform the indicated elementary row operation. 22. [52331031201318] Add 2 times Row 1 to Row 2.23E24E25E26E27E28E29E30E31ELinear Systems with One Solution The system of linear equations has a unique solution. Find the solution using Gaussian elimination or Gauss-Jordan elimination. 32. {x+y+z=14x+2y+3z=172xy=733E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53ESolving a Linear System Solve the system of linear equations. 54. {3x+y=24x+3y+z=42x+5y+z=055E56E57E58E59E60E61E62E63E64E65E66E67E68ENutrition A doctor recommends that a patient take 50 mg each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home the patient finds three brands of vitamin pills. The amounts of the relevant vitamins per pill are given in the table. How many pills of each type should he take every day to get 50 mg of each vitamin?Mixtures A chemist has three acid solutions at various concentrations. The first is 10% acid, the second is 20%, and the third is 40%. How many milliliters of each should she use to make 100 mL of 18% solution, if she has to use four times as much of the 10% solution as the 40% solution?Distance, Speed, and Time Amanda, Bryce, and Corey enter a race in which they have to run, swim, and cycle over a marked course. Their average speeds are given in the table. Corey finishes first with a total time of 1 h 45 min. Amanda comes in second with a time of 2 h 30 min. Bryce finishes last with a lime of 3 h. Find the distance (in mi) for each part of the race.Classroom Use A small school has 100 students who occupy three classrooms: A, B, and C. After the first period of the school day, half the students in room A move to room B, one-fifth of the students in room B move to room C, and one-third of the students in room C move to room A. Nevertheless, the total number of students in each room is the same for both periods. How many students occupy each room?Manufacturing Furniture A furniture factory makes wooden tables, chairs, and armoires. Each piece of furniture requires three operations: cutting the wood, assembling, and finishing. Each operation requires the number of hours given in the table. The workers in the factory can provide 300 h of cutting, 400 h of assembling, and 590 h of finishing each work week. How many tables, chairs, and armoires should be produced so that all available labor-hours are used? Or is this impossible?Traffic Flow A section of a citys street network is shown in the figure. The arrows indicate one-way streets, and the numbers show how many cars enter or leave this section of the city via the indicated street in a certain one-hour period. The variables x, y, z, and w represent the number of cars that travel along the portions of First, Second, Avocado, and Birch Streets during this period. Find x, y, z, and w, assuming that none of the cars stop or park on any of the streets shown.75EWe can add (or subtract) two matrices only if they have the same _______________.(a) We can multiply two matrices only if the number of __________ in the first matrix is the same as the number of __________ in the second matrix. (b) If A is a 3 3 matrix and B is a 4 3 matrix, which of the following matrix multiplications are possible? (i) AB (ii) BA (iii) AA (iv) BB3EFill in the missing entries in the product matrix. [312120132][132321210]=[477755]5E6EEquality of Matrices Find the values of a and b that make the matrices A and B equal. 7. A=[341a]B=[b415]8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52EEducation and Income A womens group takes a survey to determine the education and income of its members. Matrix A summarizes the proportions of members in various categories of years of postsecondary education and income. Matrix B shows the total number of members in each income category. (a) Calculate the product matrix AB. (b) Interpret the entries of the matrix AB. TotalLessthan50,00050,000to100,000100,000ormore[42010]=B54EFrozen-Food Revenue Some of the frozen foods that Joes Specialty Foods sells are pesto pizza, spinach ravioli, and macaroni and cheese. The sales distribution for these products is tabulated in matrix A. The retail price (in dollars) for each item is tabulated in matrix B. (a) Calculate the product matrix AB. (b) What is the total revenue for Monday? (c) What is the total revenue from all three days? PricePizzaRavioliMacCheese[3.505.754.25]=BFast-Food Sales A small fast-food chain with restaurants in Santa Monica, Long Beach, and Anaheim sells only hamburgers, hot dogs, and milk shakes. On a certain day, sales were distributed according to the following matrix. The price of each item is given by the following matrix. HamburgerHotdogMilkshake[0.900.801.10]=B (a) Calculate the product BA. (b) Interpret the entries in the product matrix BA.Car-Manufacturing Profits A specialty-car manufacturer has plants in Auburn. Biloxi, and Chattanooga. Three models are produced, with daily production given in the following matrix. Because of a wage increase, February profits are lower than January profits. The profit per car is tabulated by model in the following matrix. JanuaryFebruaryModelKModelRModelW[10005002000120015001000]=B (a) Calculate AB. (b) Assuming that all cars produced were sold, what was the daily profit in January from the Biloxi plant? (c) What was the total daily profit (from all three plants) in February?Canning Tomato Products Jaeger Foods produces tomato sauce and tomato paste, canned in small, medium, large, and giant-sized cans. The matrix A gives the size (in ounces) of each container. SmallMediumLargeGaintOunces[6101428]=A The matrix B tabulates one days production of tomato sauce and tomato paste. (a) Calculate the product AB. (b) Interpret the entries in the product matrix AB.Produce Sales A farmers three children, Amy, Beth, and Chad, run three roadside produce stands during the summer months. One weekend they all sell watermelons, yellow squash, and tomatoes. The matrices A and B tabulate the number of pounds of each product sold by each sibling on Saturday and Sunday. The matrix C gives the price per pound (in dollars) for each type of produce that they sell. PriceperpoundMelonsSquashTomatoes[0.100.501.00]=C Perform each of the following matrix operations, and interpret the entries in each result. (a) AC (b) BC (c) A + B (d) (A + B)CDigital Images A four-level gray scale is shown below. (a) Use the gray scale to find a 6 6 matrix that digitally represents the image in the figure. (b) Find a matrix that represents a darker version of the image in the figure. (c) The negative of an image is obtained by reversing light and dark, as in the negative of a photograph. Find the matrix that represents the negative of the image in the figure. How do you change the elements of the matrix to create the negative? (d) Increase the contrast of the image by changing each 1 to a 0 and each 2 to a 3 in the matrix you found in part (a). Draw the image represented by the resulting matrix. Does this clarify the image? (e) Draw the image represented by the matrix I. Can you recognize what this is? If you dont, try increasing the contrast. I=[123320030101132300030101133230010101]61E62EDISCOVER: Powers of a Matrix Let A=[1111] Calculate A2, A3, A4, until you detect a pattern. Write a general formula for An.DISCUSS: Square Roots of Matrices A square root of a matrix B is a matrix A with the property that A2 = B. (This is the same definition as for a square root of a number.) Find as many square roots as you can of each matrix: [4009][1509] [Hint: If A=[acbd], write the equations that a, b, c, and d would have to satisfy if A is the square root of the given matrix.](a) The matrix I=[1001] is called an __________ matrix. (b) If A is a 2 2 matrix, then A I = __________ and I A = __________. (c) If A and B are 2 2 matrices with AB = I, then B is the __________of A.2E3E4E5E6E7E8EThe Inverse of a 2 2 Matrix Use a graphing calculator to find the inverse of the matrix and to verify that A1A = AA1 = I2 and B1B = BB1 = I3. (On a TI-83, use the F r a c command to obtain the answer in fractions.) 9. A=[1.20.31.20.2]10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38ESolving a Linear System as a Matrix Equation Solve the system of equations by converting to a matrix equation and using the inverse of the coefficient matrix, as in Example 6. Use the inverses from Exercises 1114, 19, 20, 23, and 25. 39. {3x5y=42x+3y=040E41E42E43E44E45E46ESolving a Linear System Solve the system of equations by converting to a matrix equation. Use a graphing calculator to perform the necessary matrix operations, as in Example 7. 47. {x+y2z=32x+5z=112x+3y=1248E49E50E51E52E53E54EInverses of Special Matrices Find the inverse of the matrix. 55. [aaaa](a0)56E57EWhen Do Matrices Have Inverses? Find the inverse of the matrix. For what value(s) of x, if any, does the matrix have no inverse? 58. [exe2xe2xe3x]59E60ENutrition A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce. (a) Find the inverse of the matrix [313424324] and use it to solve the remaining parts of this problem. (b) How many ounces of each food should the nutritionist feed his laboratory rats if he wants their daily diet to contain 10 mg of folic acid, 14 mg of choline, and 13 mg of inositol? (c) How much of each food is needed to supply 9 mg of folic acid, 12 mg of choline, and 10 mg of inositol? (d) Will any combination of these foods supply 2 mg of folic acid, 4 mg of choline, and 11 mg of inositol?62ESales Commissions A saleswoman works at a kiosk that offers three different models of cell phones: standard with 16 GB capacity, deluxe with 32 GB capacity, and super-deluxe with 64 GB capacity. For each phone that she sells, she earns a commission based on the cell phone model. One week she sells 9 standard, 11 deluxe, and 8 super-deluxe and makes 740 in commission. The next week she sells 13 standard, 15 deluxe, and 16 super-deluxe for a S1204 commission. The third week she sells 8 standard, 7 deluxe, and 14 super-deluxe, earning 828 in commission. (a) Let x, y, and z represent the commission she earns on standard, deluxe, and super-deluxe, respectively. Translate the given information into a system of equations in x, y, and z. (b) Express the system of equations you found in part (a) as a matrix equation of the form AX = B. (c) Find the inverse of the coefficient matrix A and use it to solve the matrix equation in part (b). How much commission does the saleswoman earn on each model of cell phone?64ETrue or false? det(A) is defined only for a square matrix A.2ETrue or false? If det(A) = 0, then A is not invertible.4E5E6E7E8E9E10E11E12E13E14E15EMinors and Cofactors Evaluate the minor and cofactor using the matrix A. A=[1012352004] 16. M33, A3317E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35EDeterminants Using Row and Column Operations Evaluate the determinant, using row or column operations whenever possible to simplify your work. 36. |23174623770531240|37E38E39E40E41E42E43E44E45E46E47E48E49E50E51ECramers Rule Use Cramers Rule to solve the system. 52. {2xy=55x+3z=194y+7z=1753E54ECramers Rule Use Cramers Rule to solve the system. 55. {x+y+z+w=02x+w=0yz=0x+2z=156E57E58E59E60E61E