Chapter 9

Sketch the triangle with the given vertices and use a determinant to find its area. $$(0,0),(6,2),(3,8)$$

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 (h) given in the table. The workers in the factory can provide 300 hours of cutting, 400 hours of assembling, and 590 hours 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? $$\begin{array}{|l|ccc|} \hline & \text { Table } & \text { Chair } & \text { Armoire } \\ \hline \text { Cutting (h) } & \frac{1}{2} & 1 & 1 \\ \text { Assembling (h) } & \frac{1}{2} & 1 \frac{1}{2} & 1 \\ \text { Finishing (h) } & 1 & 1 \frac{1}{2} & 2 \end{array}$$

A researcher performs an experiment to test a hypothesis that involves the nutrients niacin and retinol. She feeds one group of laboratory rats a daily diet of precisely 32 units of niacin and \(22,000\) units of retinol. She uses two types of commercial pellet foods. Food A contains 0.12 unit of niacin and 100 units of retinol per gram. Food \(\mathrm{B}\) contains 0.20 unit of niacin and 50 units of retinol per gram. How many grams of each food does she feed this group of rats each day?

Sketch the triangle with the given vertices and use a determinant to find its area. $$(1,0),(3,5),(-2,2)$$

Making a Stovepipe \(\quad\) A rectangular piece of sheet metal with an area of 1200 in \(^{2}\) is to be bent into a cylindrical length of stovepipe having a volume of 600 in \(^{3}\). What are the dimensions of the sheet metal? GRAPH CANT COPY

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?

Sketch the triangle with the given vertices and use a determinant to find its area. $$(-1,3),(2,9),(5,-6)$$

Polynomials Determined by a Set of Points 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.

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 \(100 \mathrm{mL}\) of the first mixed 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?

Sketch the triangle with the given vertices and use a determinant to find its area. $$(-2,5),(7,2),(3,-4)$$

What must be true about the dimensions of the matrices \(A\) and \(B\) if both products \(A B\) and \(B A\) are defined?

Intersection of a Parabola and a Line On a sheet of graph paper, or using a graphing calculator, draw the parabola \(y=x^{2} .\) Then draw the graphs of the linear equation \(y=x+k\) on the same coordinate plane for various values of \(k .\) Try to choose values of \(k\) so that the line and the parabola intersect at two points for some of your \(k\) 's, and not for others. For what value of \(k\) is there exactly one intersection point? Use the results of your experiment to make a conjecture about the values of \(k\) for which the following system has two solutions, one solution, and no solution. Prove your conjecture. $$\left\\{\begin{array}{l}y=x^{2} \\\y=x+k\end{array}\right.$$

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?

$$\text { Show that }\left|\begin{array}{lll} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array}\right|=(x-y)(y-z)(z-x)$$

Let $$A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right]$$ Calculate \(A^{2}, A^{3}, A^{4}, \ldots\) until you detect a pattern. Write a general formula for \(A^{n}\).

Follow the hints and solve the systems. (a) \(\left\\{\begin{array}{cc}\log x+\log y=\frac{3}{2} & \text { [Hint: Add the equations.] } \\ 2 \log x-\log y=0 & \end{array}\right.\) (b) \(\left\\{\begin{array}{ll}2^{x}+2^{y}=10 & \text { [Hint: Note that } \\\ 4^{x}+4^{y}=68 & 4^{x}=2^{2 x}=\left(2^{x}\right)^{2}\end{array}\right]\) (c) \(\left\\{\begin{array}{cc}x-y=3 & \text { [Hint: Factor the left side } \\\ x^{3}-y^{3}=387 & \text { of the second equation. } ]\end{array}\right.\) (d) \(\left\\{\begin{array}{l}x^{2}+x y=1 \\ x y+y^{2}=3\end{array}\right.\) [Hint: Add the equations and factor the result.]

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 roadside fruit stand sells apples at \(75 \notin\) a pound, peaches at \(90 €\) a pound, and pears at \(60 \notin\) a pound. Muriel buys 18 pounds of fruit at a total cost of \(\$ 13.80 .\) Her peaches and pears together cost \(\$ 1.80\) more than her apples. (a) Set up a linear system for the number of pounds of apples, peaches, and pears that she bought. (b) Solve the system using Cramer's Rule.

Let \(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] .\) Calculate \(A^{2}, A^{3}\) \(A^{4}, \ldots\) until you detect a pattern. Write a general formula for \(A^{n}\)

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?

\(A\) square root of a matrix \(B\) is a matrix \(A\) with the property that \(A^{2}=B\). (This is the same definition as for a square root of a number.) Find as many square roots as you can of each matrix: $$\left[\begin{array}{ll} 4 & 0 \\ 0 & 9 \end{array}\right] \quad\left[\begin{array}{ll} 1 & 5 \\ 0 & 9 \end{array}\right]$$ [Hint: If \(A=$$\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) write the equations that \(a, b, c,\) and \(d\) would have to satisfy if \(A\) is the square root of the given matrix.]

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.

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 \(y=-4 x+20\).

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 Focus on Modeling (see page 240). 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 \(\sum_{k=1}^{n} x_{k}\) stands for the sum of all the \(x^{\prime}\) 's. See Section \(11.1 \text { for a complete description of sigma }(\Sigma) \text { notation. }]\) $$\left(\sum_{k=1}^{n} x_{k}\right) a+n b=\sum_{k=1}^{n} y_{k}$$ $$\left(\sum_{k=1}^{n} x_{k}^{2}\right) a+\left(\sum_{k=1}^{n} x_{k}\right) b=\sum_{k=1}^{n} x_{k} y_{k}$$ 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.

(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 cach set of points is collinear, Graph them to verify your answer. (i) \((-6,4),(2,10),(6,13)\) (ii) \((-5,10),(2,6),(15,-2)\)

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

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.