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nature of mathematics

Nature Of Mathematics 13th Edition Karl J. Smith - Solutions

Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}2 x+3 y=9 \\ x-6 y=42\end{array}ight.\)
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}x-6 y=-3 \\ 2 x+3 y=9\end{array}ight.\)
Find the corner points for the set of feasible solutions for the constraints given in Problems 35-46.\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ 10 x+5 y \geq 200 \\ 2 x+5 y \geq 100 \\ 3 x+4 y \geq 120\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x-6 y=-3 \\ 2 x+3 y=9\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}2 x+3 y=-22 \\ x-6 y=49\end{array}ight.\)
A plane makes an 870 -mile flight in \(3 \frac{1}{3}\) hours against a strong head wind, but returns in 50 minutes less with the wind. What is the plane's speed without the wind?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}y=3 x+1 \\ x-2 y=8\end{array}ight.\)
Find the corner points for the set of feasible solutions for the constraints given in Problems 35-46.\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ x+y \leq 9 \\ 2 x-3 y \geq-6 \\ x-y \leq 3\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}4 x-3 y=1 \\ 5 x+2 y=7\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}2 x+3 y=12 \\ x-6 y=-24\end{array}ight.\)
A plane with a tail wind makes its 945-mile flight in 3 hours. The return flight against the wind takes a half hour longer. What is the wind speed?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}2 x+3 y=9 \\ x=5 y-2\end{array}ight.\)
Find the corner points for the set of feasible solutions for the constraints given in Problems 35-46.\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ 2 x+y \geq 8 \\ y \leq 5 \\ x-y \leq 2 \\ 3 x-2 y \geq 5\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}3 x+7 y=5 \\ 4 x+9 y=7\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}x+2 z=7 \\ 2 x+y=16 \\ -2 y+9 z=-3\end{array}ight.\)
A dairy has cream containing \(23 \%\) butterfat and milk that is \(3 \%\) butterfat. How much of each must be mixed to obtain 30 gallons of a richer milk containing \(4 \%\) butterfat?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}x+y=4 \\ 2 x+3 y=9\end{array}ight.\)
Find the corner points for the set of feasible solutions for the constraints given in Problems 35-46.\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ 2 x+y \geq 8 \\ x-2 y \leq 7 \\ x-y \geq-3 \\ x \leq 9\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x-y=2 \\ 2 x+3 y=9\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}x+2 z=4 \\ 2 x+y=0 \\ -2 y+9 z=19\end{array}ight.\)
A chemist has two solutions of sulfuric acid. One is a \(50 \%\) solution, and the other is a \(75 \%\) solution. How many liters of each does the chemist mix to get 10 liters of a \(60 \%\) solution?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}3 x+4 y=8 \\ x+2 y=2\end{array}ight.\)
Find the optimum value for each objective function given in Problems 47-51.Maximize \(W=30 x+20 y\) subject to the constraints of Problem 35.Data from problem 35\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ 2 x+y \leq 12 \\ x+2 y \leq 9\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}3 x-4 y=16 \\ -x+2 y=-6\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}x+2 z=4 \\ 2 x+y=0 \\ -2 y+9 z=31 \\ \end{array}ight.\)
The combined height of the Transamerica Tower and the Bank of America Building is 1,632 ft. The Transamerica Tower is \(74 \mathrm{ft}\) taller. What is the height of each of the San Francisco skyscrapers?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}2 x-y=6 \\ 4 x+y=3\end{array}ight.\)
Find the optimum value for each objective function given in Problems 47-51.Maximize \(T=100 x+10 y\) subject to the constraints of Problem 36.Data from problem 36\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ 2 x+5 y \leq 20 \\ 2 x+y \leq 12\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}x+2 z=7 \\ 2 x+y=1 \\ -2 y+9 z=28\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x-y=1 \\ x+z=1 \\ y-z=1\end{array}ight.\)
The Standard Oil and the Willis (Sears) Tower have a combined height of 2,590 ft. The Sears Tower is \(318 \mathrm{ft}\) taller. What is the height of each of these Chicago towers?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}6 x+9 y=-4 \\ 9 x+3 y=1\end{array}ight.\)
Find the optimum value for each objective function given in Problems 47-51.Maximize \(P=100 x+100 y\) subject to the constraints of Problem 37.Data from problem 37\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ 3 x+2 y \leq 12 \\ x+2 y \leq 8\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x+y=2 \\ x-z=1 \\ -y+z=1\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}x+2 z=12 \\ 2 x+y=0 \\ -2 y+9 z=10\end{array}ight.\)
End to end, the Verrazano-Narrows and the George Washington bridges would span 7,760 ft. If the Verrazano-Narrows is the longer of the two New York structures by \(760 \mathrm{ft}\), how long is it?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}5 x-2 y=-1 \\ 3 x+y=17\end{array}\quadight.\)
Find the optimum value for each objective function given in Problems 47-51.Minimize \(K=6 x+18 y\) subject to the constraints of Problem 38.Data from problem 38\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ x \leq 10 \\ y \leq 8 \\ 3 x+2 y \geq 12\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x+5 z=9 \\ y+2 z=2 \\ 2 x+3 z=4\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}x+2 z=5 \\ 2 x+y=8 \\ -2 y+9 z=9\end{array}ight.\)
The combined length of the Golden Gate and San Francisco Bay bridges is \(6,510 \mathrm{ft}\). If the Golden Gate is \(1,890 \mathrm{ft}\) longer, what is its length?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}5 x+4 y=9 \\ 9 x+3 y=12\end{array}ight.\)
Find the optimum value for each objective function given in Problems 47-51.Minimize \(A=2 x-3 y\) subject to the constraints of Problem 39.Data from problem 39\(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 \\ x+y \leq 8 \\ y \leq 4 \\ x \leq 6\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x+2 z=13 \\ 2 x+y=8 \\ -2 y+9 z=41\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}6 x+y+20 z=27 \\ x-y=0 \\ y+3 z=4\end{array}ight.\)
Noxin Electronics has investigated the feasibility of introducing a new line of magnetic tape. The study shows that both supply and demand are linear. The supply can increase from 1,000 items at \(\$ 2\) each to 5,000 units at \(\$ 4\) each. The demand ranges from 1,000 items at \(\$ 4\) to 7,000
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}100 x-y=0 \\ 50 x+y=300\end{array}ight.\)
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}x=\frac{3}{4} y-2 \\ 3 y-4 x=5\end{array}ight.\)
The Wadsworth Widget Company manufactures two types of widgets: regular and deluxe. Each widget is produced at a station consisting of a machine and a person who finishes each widget by hand. The regular widget requires \(3 \mathrm{hr}\) of machine time and \(2 \mathrm{hr}\) of finishing time. The
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}4 x+y=-2 \\ 3 x+2 z=-9\\ 2 y+3 z=-5\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}6 x+y+20 z=14 \\ x-y=1 \\ y+3 z=1\end{array}ight.\)
Noxin Electronics is considering producing small SD cards. Research shows linear demand to be from 40,000 SD cards at \(\$ 2\) to 100,000 at \(\$ 1\). Similarly, the supply goes from 20,000 SD cards at \(50 otin\) to 80,000 at \(\$ 5\). What is the equilibrium point?
A convalescent hospital wishes to provide, at a minimum cost, a diet that has a minimum of \(200 \mathrm{~g}\) of carbohydrates, \(100 \mathrm{~g}\) of protein, and \(20 \mathrm{~g}\) of fat per day. These requirements can be met with two foods, A and B:If food A costs \(\$0.29\) per gram and food
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}q+d=147 \\ 0.25 q+0.10 d=24.15\end{array}ight.\)
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}5 x+z=9 \\ x-5 z=7 \\ x+y-z=0\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}6 x+y+20 z=-12 \\ x-y=6 \\ y+3 z=-7 \\\end{array}ight.\)
Sterling silver contains \(92.5 \%\) silver. How many grams of pure silver and sterling silver must be mixed to get 100 grams of a \(94 \%\) alloy?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}x+y=10 \\ 0.4 x+0.9 y=0.5(10)\end{array}ight.\)
Karlin Enterprises manufactures two games. Standing orders require that at least 24,000 space-battle games and 5,000 football games be produced per month. The Gainesville plant can produce 600 space-battle games and 100 football games per day; the Sacramento plant can produce 300 space-battle games
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x+y=-2 \\ y+z=2 \\ x-y-z=-1\end{array}ight.\)
Solve the systems in Problems 27-54 by solving the corresponding matrix equation with an inverse, if possible.\(\left\{\begin{array}{l}6 x+y+20 z=57 \\ x-y=1 \\ y+3 z=5\end{array}ight.\)
A pain remedy contains \(12 \%\) aspirin, and a stronger formula has \(25 \%\) aspirin, but is otherwise the same. A chemist mixes some of each to obtain \(100 \mathrm{mg}\) of a mixture with \(20 \%\) aspirin. How much of each is used?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}12 x-5 y=-39 \\ y=2 x+9\end{array}ight.\)
Write a linear programming model, including the objective function and the set of constraints, for Problems 52-55. DO NOT SOLVE, but be sure to define all your variables.Brown Bros., Inc. is an investment company doing an analysis of the pension fund for a certain company. The fund has a maximum of
If we take the first few rows of Pascal's triangle and arrange them into a lower triangular matrix, we form what is called a Pascal matrix:a. Find the inverse of this matrix.b. Make a general statement about the inverse of any order Pascal's matrix. [P] 1 0 0 0 0 1 1 0 0 0 1210 1 3 3 1 1464 0 0 1
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x+y=-1 \\ y+z=-1 \\ x+y+z=1\end{array}ight.\)
How many gallons of \(24 \%\) butterfat cream must be mixed with 500 gallons of \(3 \%\) butterfat milk to obtain a \(4 \%\) butterfat milk?
Solve the systems in Problems 39-56 for all real solutions, using any suitable method.\(\left\{\begin{array}{l}y=2 x-1 \\ y=-3 x-9\end{array}ight.\)
Solve the linear programming problems in Problems 56-59.Suppose the net profit per bushel of corn in Example 3 increased to \(\$ 2.00\) and the net profit per bushel of wheat dropped to \(\$ 1.50\). Maximize the profit if the other conditions in the example remain the same.Data from Example 3A
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x+2 z=9 \\ 2 x+y=13 \\ 2 y+z=8\end{array}ight.\)
The radiator of a car holds 17 quarts of liquid. If it now contains \(15 \%\) antifreeze, how many quarts must be replaced by antifreeze to give the car a \(60 \%\) solution in its radiator?
If we add Pascal’s matrix (Problem 55) and the identity matrix, we find [P] + [I] . For the order shown in Problem 55,Find the inverse of this matrix.Data from Problem 55If we take the first few rows of Pascal’s triangle and arrange them into a lower triangular matrix, we form what is called a
Solve the linear programming problems in Problems 56-59.Suppose the farmer in Example 3 contracted to have the grain stored at a neighboring farm and the contract calls for at least 4,000 bushels to be stored. How many acres should be planted in corn and how many in wheat to maximize profit if the
The premise of this problem is that since dogs supposedly age seven times as quickly as humans, at some point the dog will become "older" than its owner. Karen wanted to determine exactly on which day this milestone would occur for her and her dog, Sydney, so that they could celebrate the occasion.
Solve the systems in Problems \(40-57\) by the Gauss-Jordan method.\(\left\{\begin{array}{l}x+2 z=0 \\ 3 x-y+2 z=0 \quad \\ 4 x+y=6\end{array}ight.\)
How many channels of communication are open to each country in Example 4 if the countries are willing to speak to each other through two intermediaries? United States has diplomatic relations with Russia and with Mexico, but not with Cuba. Mexico has diplomatic relations with the United States and
A party punch bowl contains 5 quarts containing 20\% 7-UP, and the host wants to change it to a \(30 \%\) 7-UP solution. How much of the punch must be replaced with 7-UP to accomplish this task?
Assume that Sydney in Problem 57 is a cat instead of a dog, and assume that cats age four times as quickly as humans. Using this assumption, when will Karen and Sydney be "the same age"?Data from Problem 57The premise of this problem is that since dogs supposedly age seven times as quickly as
Solve the linear programming problems in Problems 56-59.The Thompson Company manufactures two industrial products, standard ( \(\$ 45\) profit per item) and economy (\$30 profit per item). These items are built using machine time and manual labor. The standard product requires \(3 \mathrm{hr}\) of
Consider the map of airline routes shown in Figure 16.5.Figure 16.5a. Fill in the blanks in the following zero-one matrix representing the airline routes:b. Write a matrix showing the number of routes among these cities if you make exactly one intermediate stop. Use this matrix to state in how
A winery has a large amount of a wine labeled "Lot I" that is a mixture of \(92 \%\) Merlot wine and \(8 \%\) Cabernet Sauvignon wine. It also has a second wine labeled "Lot II" that is a mixture of \(88 \%\) Cabernet Sauvignon and \(12 \%\) Merlot.The wine master decides to mix together these two
The supply curve for a certain commodity is \(n=2,500 p-500\), and the demand curve for the same product is \(n=31,500-1,500 p\), where \(n\) is the number of items and \(p\) is the number of dollars.a. At \(\$ 15\) per unit of the commodity, how many items would be supplied? How many would be
The Louvre Tablet from the Babylonian civilization is dated about 1500 B.C. It shows a system equivalent to\[\left\{\begin{array}{l}x y=1 \\x+y=a\end{array}ight.\]Solve this system for \(x\) and \(y\) in terms of \(a\). This is not a linear system, and you will need to use the quadratic formula
What is wrong, if anything, with the following statement and the provided answer? Explain your reasoning. You have identical cups, one containing coffee and one containing cream. One teaspoon of the cream is added to the coffee and stirred in. Now a teaspoon of the coffee/cream mixture is added
Solve the linear programming problems in Problems 56-59.Your broker tells you of two investments she thinks are worthwhile. She advises that you buy a new issue of Pertec stock, which should yield \(20 \%\) over the next year, and then to balance your account she advises Campbell Municipal Bonds
The following carbohydrate information is given on the side of the respective cereal boxes (for \(1 \mathrm{oz}\) of cereal with \(\frac{1}{2}\) cup of whole milk):What is the minimum cost to receive at least \(322 \mathrm{~g}\) starch and \(119 \mathrm{~g}\) sucrose by consuming these two cereals
To control a certain type of crop disease, it is necessary to use \(23 \mathrm{gal}\) of chemical A and \(34 \mathrm{gal}\) of chemical B. The dealer can order commercial Spray I, each container of which holds 5 gal of chemical A and 2 gal of chemical B, and commercial Spray II, each container of
The Seedy Vin Company produces Riesling, Charbono, and Rosé wines. There are three procedures for producing each wine (the procedures for production affect the cost of the final product). One procedure allows an outside company to bottle the wine; a second allows the wine to be produced and
The supply curve for a new software product is given by \(n=2.5 p-500\), and the demand curve for the same product is \(n=200-0.5 p\), where \(n\) is the number of items and \(p\) is the number of dollars.a. At \(\$ 250\) for the product, how many items would be supplied? How many would be
The following problem was written by Leonhard Euler: "Two persons owe conjointly 29 pistoles; they both have money, but neither of them enough to enable him, singly, to discharge this common debt." The first debtor says therefore to the second, "If you give me \(\frac{2}{3}\) of your money, I can
A taste test is conducted on the Atlantic City Boardwalk. People are given samples of Coke, Pepsi, and Safeway brands of cola in unmarked cups, and are then asked to rank them in order of preference. The first cup is labeled A, the second, \(\mathrm{B}\), and the third, \(\mathrm{C}\). Here are the
A candy maker mixes chocolate, milk, and mint extract to produce three kinds of candy (I, II, and III) with the following proportions:I: \(7 \mathrm{lb}\) chocolate, 5 gal milk, \(1 \mathrm{oz}\) mint extract II: \(3 \mathrm{lb}\) chocolate, 2 gal milk, \(2 \mathrm{oz}\) mint extract III: \(4
Sociologists often study the dominance of one group over another. Suppose in a certain society there are four classes which we will call: abigweel, aweel, upancomer, and pon. Hint: These are pronounced "A big wheel," "A wheel," "Up and comer," and "pea-on." Abigweel dominates aweel, upancomer, and
Describe and discuss the plurality voting method.
Discuss the idea of apportionment. Describe some different apportionment schemes.
What is the Alabama paradox?
A taste test is conducted on the Atlantic City Boardwalk. People are given samples of Coke, Pepsi, and Safeway brands of cola in unmarked cups, and are then asked to rank them in order of preference. The first cup is labeled A, the second, \(\mathrm{B}\), and the third, \(\mathrm{C}\). Here are the
What is the majority criterion?
Describe and discuss the Borda count voting method.
Discuss the ideas of arithmetic mean and geometric mean.

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