Chapter 7

MODELING Predator-Prey Relationship In certain parts of the Rocky Mountains, deer are the main food source for mountain lions. When the deer population \(d\) is large, the mountain lions \((m)\) thrive. However, a large mountain lion population drives down the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$\left[\begin{array}{c}m_{n+1} \\\ d_{n+1}\end{array}\right]=\left[\begin{array}{rr}0.51 & 0.4 \\ -0.05 & 1.05\end{array}\right]\left[\begin{array}{l}m_{n} \\\ d_{n}\end{array}\right]$$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 year? 2 years? (c) Consider part (b), but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of \(1 \%\).

Each set of data in Exercises \(79-82\) can be modeled by $$f(x)=a x^{2}+b x+c$$ (a) Find a linear system whose solution represents values of \(a, b, a n d c\) (b) Find \(f(x)\) by using a method from this section. (c) Graph \(f\) and the data in the same viewing window. (d) Make your own prediction using \(f .\) Answers will vary. Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) is a greenhouse gas. Its concentration in parts per million (ppm) has been measured at Mauna Loa, Hawaii, during past years. The table lists measurements for three selected years \(x\) $$\begin{array}{l|c|c|c} \hline \text { Year } & 1958 & 1973 & 2016 \\ \hline \mathrm{CO}_{2}(\mathrm{ppm}) & 315 & 330 & 404 \end{array}$$

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned}&y \geq 2^{x}\\\&y \leq 8\end{aligned}$$

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+B=B+A\) (commutative property)

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$y \leq x^{3}+x^{2}-4 x-4$$

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+(B+C)=(A+B)+C\) (associative property)

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$3 x+2 y \geq 6$$

Solve each problem. To model spring fawn count \(F\) from adult antelope population \(A\), precipitation \(P,\) and severity of winter \(W\), environmentalists have used the equation $$F=a+b A+c P+d W$$ where the coefficients \(a, b, c,\) and \(d\) are constants that must be determined before using the equation. The table lists the results of four different (representative) years. $$\begin{array}{|c|c|c|c|} \hline \text { Fawns } & \text { Adults } & \begin{array}{c} \text { Precip. } \\ \text { (in inches) } \end{array} & \begin{array}{c} \text { Winter } \\ \text { Severity } \end{array} \\ \hline 239 & 871 & 11.5 & 3 \\ 234 & 847 & 12.2 & 2 \\ 192 & 685 & 10.6 & 5 \\ 343 & 969 & 14.2 & 1 \end{array}$$ (a) Substitute the values for \(F, A, P,\) and \(W\) from the table for each of the four years into the given equation \(F=a+b A+c P+d W\) to obtain four linear equations involving \(a, b, c,\) and \(d\) (b) Write a \(4 \times 5\) augmented matrix representing the system, and solve for \(a, b, c,\) and \(d\) (c) Write the equation for \(F,\) using the values from part (b) for the coefficients. (d) If a winter has severity \(3,\) adult antelope population \(960,\) and precipitation 12.6 inches, predict the spring fawn count. (Compare this with the actual count of \(320 .\) )

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((A B) C=A(B C)\) (associative property)

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$y \leq x^{2}+5$$

Solve each problem. The table shows weight \(W,\) neck size \(N,\) overall length \(L,\) and chest size \(C\) for four bears. $$\begin{array}{|c|c|c|c|} \hline W \text { (pounds) } & N \text { (inches) } & L \text { (inches) } & C \text { (inches) } \\ \hline 125 & 19 & 57.5 & 32 \\ 316 & 26 & 65 & 42 \\ 436 & 30 & 72 & 48 \\ 514 & 30.5 & 75 & 54 \end{array}$$ (a) We can model these data with the equation $$W=a+b N+c L+d C$$ where \(a, b, c,\) and \(d\) are constants. To do so, represent a system of linear equations by a \(4 \times 5\) augmented matrix whose solution gives values for \(a, b, c,\) and \(d\) (b) Solve the system. Round each value to the nearest thousandth. (c) Predict the weight of a bear with \(N=24, L=63,\) and \(C=39 .\) Interpret the result.

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A(B+C)=A B+A C\) (distributive property)

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$x+y \geq 2$$ $$x+y \leq 6$$

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(c(A+B)=c A+c B,\) for any real number \(c\).

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned}&y \geq|x+2|\\\&y \leq 6\end{aligned}$$

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((c+d) A=c A+d A,\) for any real numbers \(c\) and \(d\).

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. $$(c A) d=(c d) A$$

Online Black Friday Spending The total online spending on Black Friday during 2014 and 2015 was \(\$ 4437\) million. From 2014 to \(2015,\) spending increased by \(\$ 1427\) million. (a) Write a system of equations whose solution represents the Black Friday spending in each of these years. Let \(x\) be the amount spent in 2015 and \(y\) be the amount spent in 2014 (b) Solve the system. (c) Interpret the solution.

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. $$(c d) \mathcal{A}=c(d A)$$

Self-Reported Spending The average of self-reported spending "yesterday" for high-income consumers and middle-/low-income consumers was \(\$ 119.00\) in March \(2017 .\) High-income consumers spend \(\$ 88\) more than middle-/low- income consumers. (a) Write a system of equations whose solution gives the self-reported spending for each income group. Let \(x\) be the spending by high-income consumers and \(y\) be the spending by middle- Mow -income consumers. (b) Solve the system. (c) Interpret the solution.

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$y=2 x-1$$ $$y=2-x^{2}$$

Price of Smartphones From 2014 to \(2016,\) the average selling price of smartphones decreased by \(10 \% .\) This percent reduction amounted in a decrease of \(\$ 31 .\) Find the average selling price of smartphones in 2014 and in 2016 .

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$y=x^{2}-x+1$$ $$y=-x^{2}+1$$

Dimensions of a Box A box has an open top, rectangular sides, and a square base. The volume of the box is 576 cubic inches, and the surface area of the outside of the box is 336 square inches. Find the dimensions of the box.

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned}&y=x^{3}\\\&y=x\end{aligned}$$

Dimensions of a Box \(\quad\) A box has rectangular sides and a rectangular top and base that are twice as long as they are wide. The volume of the box is 588 cubic inches, and the surface area of the outside of the box is 448 square inches. Find the dimensions of the box.

Shade the region(s) contained inside the graphs and give any points of intersection of the equations. $$\begin{aligned}&y=2 x^{2}+x-3\\\&y=x^{2}-2 x+1\end{aligned}$$

Investments \(\quad\) A student invests a total of \(\$ 5000\) at \(3 \$ 6\) and \(4 \%\) annually. After 1 year, the student receives a total of \(\$ 187.50\) in interest. How much did the student invest at each interest rate?

Find a system of linear inequalities for which the graph is the region in the first quadrant between and inclusive of the pair of lines \(x+2 y-8=0\) and \(x+2 y=12\).

Investments \(A\) student invests a total of \(\$ 7000\) at \(1.5 \%\) and \(2 \%\) annually. After 1 year, the student receives a total of \(\$ 128.50\) in interest. How much did the student invest at each interest rate?

Advertising Spending In \(2015,\) AT\&T and Verizon spent a combined \(\$ 1212\) million on network TV advertising. AT\&T spent \(\$ 250\) million more than Verizon. (a) Write a system of equations whose solution gives the spending of each company, in millions of dollars. Let \(x\) be the amount spent by AT\&T and \(y\) be the amount spent by Verizon. (b) Solve the system of equations. (c) Interpret the solution.

Populations of Minorities in the United States \(\quad\) The current and estimated resident populations, \(y\) (in percent), of Black and Spanish/Hispanic/Latino people in the United States for the years \(1990-2050\) are modeled by the following linear equations. $$ \begin{array}{ll} y=0.0515 x+12.3 & \text { Black } \\ y=0.255 x+9.01 & \text { SpM } \end{array} $$ Sp.Hisp \(\Omega\) at. In each case, \(x\) represents the number of years after 1990 . (a) Solve the system to find the year when these population percents were equal. (b) What percent of the U.S. resident population was Spanish/Hispanic/Latino in the year found in part (a)? (c) Graphically support the analytic solution in part (a). (d) Which population is increasing more rapidly?

Geometry Determine graphically whether it is possible to construct a cylindrical container, including the top and bottom, with volume 38 cubic inches and surface area 38 square inches.

Solve each problem..Inquiries about Displayed Products \(\quad\) A wholesaler of party goods wishes to display her products at a convention of social secretaries in such a way that she gets the maximum number of inquiries about her whistles and hats. Her booth at the convention has 12 square meters of floor space to be used for display purposes. A display unit for hats requires 2 square meters, and one for whistles requires 4 square meters. Experience tells the wholesaler that she should never have more than a total of 5 units of whistles and hats on display at one time. If she receives three inquiries for each unit of hats and two inquiries for each unit of whistles on display, how many of each should she display in order to get the maximum number of inquiries? What is that maximum number?

Profit from Farm Animals Farmer Jones raises only pigs and geese. She wants to raise no more than 16 animals, with no more than 12 geese. She spends \(\$ 50\) to raise a pig and \(\$ 20\) to raise a goose. She has \(\$ 500\) available for this purpose. Find the maximum profit she can make if she makes a profit of \(\$ 80\) per goose and \(\$ 40\) per pig. Indicate how many pigs and geese she should raise to achieve this maximum.

Height and Weight The relationship between a professional basketball player's height \(h\) in inches and weight \(w\) in pounds was modeled by using two samples of players. The resulting equations were $$ w=7.46 h-374 $$ $$ \text { and } \quad w=7.93 h-405 $$ Assume that \(65 \leq h \leq 85\) (a) Use each equation to predict the weight to the nearest pound of a professional basketball player who is 6 feet 11 inches tall. (b) Determine graphically the height at which the two models give the same weight. (c) For each model, what change in weight is associated with a 1 -inch increase in height?

Shipment Costs A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25 refrigerators already, while warehouse \(B\) has 20 on hand. It costs \(\$ 12\) to ship a refrigerator to warehouse \(A\) and \(\$ 10\) to ship one to warehouse B. How many refrigerators should be shipped to each warehouse to minimize cost? What is the minimum cost?

Theo requires two food supplements:I and II. He can get these supplements from two different products \(A\) and \(B,\) as shown in the following table.$$\begin{array}{l|c|c}\hline \text { Supplement (grams/serving) } & \text { I } & \text { II } \\\\\hline \text { Product } A & 3 & 2 \\\\\text { Product } B & 2 & 4\end{array}$$.Theo's physician recommends at least 15 grams of each supplement in Theo's daily diet. If product \(A\) costs \(25 \phi\) per serving and product \(B\) costs \(40 \notin\) per serving, how can he satisfy his requirements most economically?

The supply of a certain product is related to its price by the equation \(p=\frac{1}{3} q,\) where \(p\) is in dollars and \(q\) is the quantity supplied in hundreds of units. (a) If this product sells for \(\$ 9,\) what quantity will be supplied by the manufacturer? (b) Suppose that consumer demand for the same product decreases as price increases according to the equation \(p=20-\frac{1}{5} q\). If this product sells for \(\$ 9,\) what quantity will consumers purchase? How does this compare with the quantity being supplied by the manufacturer at this price? (c) On the basis of parts (a) and (b), what should happen to the price? Explain. (d) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?

A manufacturing process requires that oil refineries manufacture at least 2 gallons of gasoline for each gallon of fuel oil. To meet winter demand for fuel oil, at least 3 million gallons a day must be produced. The demand for gasoline is no more than 6.4 million gallons per day. If the price of gasoline is \(\$ 1.90\) per gallon and the price of fuel oil is \(\$ 1.50\) per gal. Ion, how much of each should be produced to maximize revenue?

Suppose that supply is related to price by \(p=\frac{1}{10} q\) and that demand is related to price by \(p=15-\frac{2}{3} q,\) where \(p\) is price in dollars and \(q\) is the quantity supplied in units. (a) Determine the price at which 15 units would be supplied. Determine the price at which 15 units would be demanded. (b) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?

A shop manufactures two types of bolts on three groups of machines. The time required on each group differs, as shown in the following table.\begin{array}{|c|c|c|c|}\hline\hline & \text { Machine Group } \\ \hline \text { Bolt } & \mathbf{I} & \mathbf{I I} & \mathbf{I I I} \\\\\hline \text { Type A } & 0.1 \mathrm{min} & 0.1 \mathrm{min} & 0.1 \mathrm{min} \\\\\text { Type B } & 0.1 \mathrm{min} & 0.4 \mathrm{min} & 0.5 \mathrm{min}\end{array}.In a day, there are \(240,720,\) and 160 minutes available, respectively, on these machines. Type A bolts sell for S0.10 and Type B bolts for S0.12. How many of each type of bolt should be manufactured per day to maximize revenue? What is the maximum revenue?

Find the equilibrium price in dollars if$$p=\frac{2}{3} q \quad \text { and } \quad p=49-\frac{1}{2} q$$ How many units represent the demand at this price?

Earthquake victims need medical supplies and bottled water. Each medical kit measures 1 cubic foot and weighs 10 pounds. Each container of water is also 1 cubic foot, but weighs 20 pounds. The plane can carry only \(80,000\) pounds, with total volume 6000 cubic feet. Each medical kit will aid 4 people, while each container of water will serve 10 people. How many of each should be sent in order to maximize the number of people aided? How many people will be aided?

Point The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise. C represents cost in dollars to produce x items. and \(R\) represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R\). Then find the value of \(C\) and \(R\) at that point. \(C=20 x+10,000\) \(R=30 x-11,000\)

Point The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise. C represents cost in dollars to produce x items. and \(R\) represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R\). Then find the value of \(C\) and \(R\) at that point. \(c=4 x+125\) \(R=9 x-200\)