Chapter 11

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?

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

Number Problem The sum of two numbers is twice their difference. The larger number is 6 more than twice the smaller. Find the numbers.

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

Value of Coins A man has 14 coins in his pocket, all of which are dimes and quarters. If the total value of his change is S2. \(75,\) how many dimes and how many quarters does he have?

Buying Fruit A roadside fruit stand sells apples at 75 \(\mathrm{c}\) a pound, peaches at 90 \(\mathrm{c}\) a pound, and pears at 60 \(\mathrm{c}\) 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.

Admission Fees The admission fee at an amusement park is \(\$ 1.50\) for children and \(\$ 4.00\) for adults. On a certain day, 2200 people entered the park, and the admission fees that were collected totaled \(\$ 5050 .\) How many children and how many adults were admitted?

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. $$ (0,0),(1,12), \quad(2,40), \quad(3,6), \quad(-1,-14) $$ Graph the points and functions in the same viewing rectangle using a graphing device.

Gas Station A gas station sells regular gas for \(\$ 2.20\) per gallon and premium gas for \(\$ 3.00\) allon. At the end of a business day 280 gallons of gas were sold, and receipts totaled S680. How many gallons of each type of gas were sold?

Fruit Stand A fruit stand sells two varieties of strawberries: standard and deluxe. A box of standard strawberries sells for \(\$ 7,\) and a box of deluxe strawberries sells for \(\$ 10 .\) In one day the stand sells 135 boxes of strawberries for a total of \(\$ 1110 .\) How many boxes of each type were sold?

Determinant Formula for the Area of a Triangle The figure shows a triangle in the plane with vertices \(\left(a_{1}, b_{1}\right)\) , \(\left(a_{2}, b_{2}\right),\) and \(\left(a_{3}, b_{3}\right)\) (a) Find the coordinates of the vertices of the surrounding rectangle, and find its area. (b) Find the area of the red triangle by subtracting the areas of the three blue triangles from the area of the rectangle. (c) Use your answer to part (b) to show that the area of the red triangle is given by

Airplane Speed \(A\) man flies a small airplane from Fargo to Bismarck, North Dakota-a distance of 180 mi. Because he is flying into a head wind, the trip takes him 2 hours. On the way back, the wind is still blowing at the same speed, so the return trip takes only 1 \(\mathrm{h} 12\) min. What is his speed in still air, and how fast is the wind blowing?

Collinear Points and Determinants (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) \((-5,10),(2,6),(15,-2)\)

Boat Speed A boat on a river travels downstream between two points, 20 mi apart, in one hour. The return trip against the current takes 2\(\frac{1}{2}\) hours. What is the boat's speed, and how fast does the current in the river flow?

Nutrition 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 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?

Matrices with Determinant Zero 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

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?

Solving Linear Systems 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.

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, 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?

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

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?

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 be- cause it is less risky. His annual interest is \(\$ 3520 .\) How much did he invest at each rate?

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 \(\mathrm{min}\) longer than John's. For what length of time does each of them drive?

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

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.

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 2 (see page \(171 ) .\) By using calculus, it can be shown that the line that best fits the \(n\) data points \(\left(x_{1}, = y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{m} y_{n}\right)\) 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 \(X^{\prime}\) see Section 13.1 for a complete description of sigma ( \(\Sigma\) ) 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.