Chapter 10

Coffee Blends \(A\) customer in a coffee shop purchases a blend of two coffees: Kenyan, costing \(\$ 3.50\) a pound, and Sri Lankan, costing \(\$ 5.60\) a pound. He buys 3 lb of the blend, which costs him \(\$ 11.55 .\) How many pounds of each kind went into the mixture?

Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$\left\\{\begin{array}{l} y \geq x^{3} \\ 2 x+y \geq 0 \\ y \leq 2 x+6 \end{array}\right.$$

(a) If three points lie on a line, what is the area of the "triangle" that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points \(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right),\) and \(\left(a_{3}, b_{3}\right)\) are collinear if and only if $$ \left|\begin{array}{lll} a_{1} & b_{1} & 1 \\ a_{2} & b_{2} & 1 \\ a_{3} & b_{3} & 1 \end{array}\right|=0 $$ (b) Use a determinant to check whether each set of points is collinear. Graph them to verify your answer. (i) \((-6,4),(2,10),(6,13)\) (ii) \((-5,10),(2,6),(15,-2)\)

Mixture Problem A chemist has two large containers of sulfuric acid solution, with different concentrations of acid in each container. Blending \(300 \mathrm{mL}\) of the first solution and \(600 \mathrm{mL}\) of the second gives a mixture that is \(15 \%\) acid, whereas blending \(100 \mathrm{mL}\) of the first with \(500 \mathrm{mL}\) of the second gives a \(12 \frac{1}{2} \%\) acid mixture. What are the concentrations of sulfuric acid in the original containers?

A farmer has 500 acres of arable land on which he wants to plant potatoes and corn. The farmer has \(\$ 40,000\) available for planting and \(\$ 30,000\) for fertilizer. Planting 1 acre of potatoes costs \(\$ 90,\) and planting 1 acre of corn costs \(\$ 50 .\) Fertilizer costs \(\$ 30\) for 1 acre of potatoes and \(\$ 80\) for 1 acre of com. (a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region. (b) Can the farmer plant 300 acres of potatoes and 180 acres of corn? (c) Can the farmer plant 150 acres of potatoes and 325 acres of corn?

Mixture Problem \(A\) biologist has two brine solutions, one containing \(5 \%\) salt and another containing \(20 \%\) salt. How many milliliters of each solution should she mix to obtain \(1 \mathrm{L}\) of a solution that contains \(14 \%\) salt?

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 \(100 \mathrm{mL}\) of \(18 \%\) solution, if she has to use four times as much of the \(10 \%\) solution as the \(40 \%\) solution?

A farmer has 300 acres of arable land on which she wants to plant cauliflower and cabbage. The farmer has \(\$ 17,500\) available for planting and \(\$ 12,000\) for fertilizer. Planting 1 acre of cauliflower costs \(\$ 70,\) and planting 1 acre of cabbage costs \(\$ 35 .\) Fertilizer costs \(\$ 25\) for 1 acre of cauliflower and \(\$ 55\) for 1 acre of cabbage. (a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region. (b) Can the farmer plant 155 acres of cauliflower and 115 acres of cabbage? (c) Can the farmer plant 115 acres of cauliflower and 175 acres of cabbage?

Investments \(A\) woman invests a total of \(\$ 20,000\) in two accounts, one paying \(5 \%\) and the other paying \(8 \%\) simple interest per year. Her annual interest is \(\$ 1180 .\) How much did she invest at each rate?

Publishing Books A publishing company publishes a total of no more than 100 books every year. At least 20 of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.

Investments A man invests his savings in two accounts, one paying \(6 \%\) and the other paying \(10 \%\) simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His annual interest is \(\$ 3520 .\) How much did he invest at each rate?

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?

Furniture Manufacturing A man and his daughter manufacture unfinished tables and chairs. Each table requires 3 h of sawing and 1 h of assembly. Each chair requires 2 h of sawing and \(2 \mathrm{h}\) of assembly. Between the two of them, they can put in up to 12 h of sawing and 8 h of assembly work each day. Find a system of inequalities that describes all possible combinations of tables and chairs that they can make daily. Graph the solution set.

A coffee merchant sells two different coffee blends. The Standard blend uses 4 oz of arabica and 12 oz of robusta beans per package; the Deluxe blend uses 10 oz of arabica and 6 oz of robusta beans per package. The merchant has 80lb of arabica and 90 lb of robusta beans available. Find a system of inequalities that describes the possible number of Standard and Deluxe packages the merchant can make. Graph the solution set.

Distance, Speed, and Time John and Mary leave their house at the same time and drive in opposite directions. John drives at \(60 \mathrm{mi} / \mathrm{h}\) and travels \(35 \mathrm{mi}\) farther than Mary, who drives at \(40 \mathrm{mi} / \mathrm{h}\). Mary's trip takes 15 min longer than John's. For what length of time does each of them drive?

A cat food manufacturer uses fish and beef byproducts. The fish contains 12 g of protein and 3 g of fat per ounce. The beef contains 6 g of protein and 9 g of fat per ounce. Each can of cat food must contain at least \(60 \mathrm{g}\) of protein and 45 g of fat. Find a system of inequalities that describes the possible number of ounces of fish and beef byproducts that can be used in each can to satisfy these minimum requirements. Graph the solution set.

Aerobic Exercise \(\quad\) A woman keeps fit by bicycling and running every day. On Monday she spends \(\frac{1}{2} \mathrm{h}\) at each activity, covering a total of \(12 \frac{1}{2} \mathrm{mi}\). On Tuesday she runs for 12 min and cycles for 45 min, covering a total of 16 mi. Assuming that her running and cycling speeds don't change from day to day, find these speeds.

We all know that two points uniquely determine a line \(y=a x+b\) in the coordinate plane. Similarly, three points uniquely determine a quadratic (second-degree) polynomial $$ y=a x^{2}+b x+c $$ four points uniquely determine a cubic (third-degree) polynomial $$ y=a x^{3}+b x^{2}+c x+d $$ and so on. (Some exceptions to this rule are if the three points actually lie on a line, or the four points lie on a quadratic or line, and so on.) For the following set of five points, find the line that contains the first two points, the quadratic that contains the first three points, the cubic that contains the first four points, and the fourth-degree polynomial that contains all five points. \((0,0), \quad(1,12), \quad(2,40), \quad(3,6), \quad(-1,-14)\) Graph the points and functions in the same viewing rectangle using a graphing device.

DISCUSS: Shading Unwanted Regions To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? Why? $$\left\\{\begin{array}{r} x+2 y>4 \\ -x+y<1 \\ x+3 y<9 \\ x<3 \end{array}\right.$$

Use the definition of determinant and the elementary row and column operations to explain why matrices of the following types have determinant 0 (a) A matrix with a row or column consisting entirely of zeros (b) A matrix with two rows the same or two columns the same (c) A matrix in which one row is a multiple of another row, or one column is a multiple of another column

Number Problem The sum of the digits of a two-digit number is \(7 .\) When the digits are reversed, the number is increased by \(27 .\) Find the number.

Suppose you have to solve a linear system with five equations and five variables without the assistance of a calculator or computer. Which method would you prefer: Cramer's Rule or Gaussian elimination? Write a short paragraph explaining the reasons for your answer.

Area of a Triangle Find the area of the triangle that lies in the first quadrant (with its base on the \(x\) -axis) and that is bounded by the lines \(y=2 x-4\) and \(y=-4 x+20\).

DISCUSS: The Least Squares Line The least squares line or regression line is the line that best fits a set of points in the plane. We studied this line in the Focus on Modeling that follows Chapter 1 (see page 139 ). By using calculus, it can be shown that the line that best fits the \(n\) data points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) is the line \(y=a x+b,\) where the coefficients \(a\) and \(b\) satisfy the following pair of linear equations. (The notation \(\Sigma_{k-1}^{n} x_{k}\) stands for the sum of all the \(x^{\prime}\) s. See Section 12.1 for a complete description of sigma \((\Sigma)\) notation.) $$\begin{array}{c} \left(\sum_{k=1}^{n} x_{k}\right) a+n b=\sum_{k=1}^{n} y_{k} \\ \left(\sum_{k=1}^{n} x_{k}^{2}\right) a+\left(\sum_{k=1}^{n} x_{k}\right) b=\sum_{k=1}^{n} x_{k} y_{k} \end{array}$$ Use these equations to find the least squares line for the following data points. \((1,3), \quad(2,5), \quad(3,6), \quad(5,6), \quad(7,9)\) Sketch the points and your line to confirm that the line fits these points well. If your calculator computes regression lines, see whether it gives you the same line as the formulas.