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Algebra And Trigonometry 10th Edition Ron Larson - Solutions

Use a graphing utility to graph the solution set of the system of inequalities.1.2. 3. 4. 5. 6.
Sketch the graph of the inequality. 1. y < 5 − x2 2. y2 − x < 0 3. x ≤ 6
Write a system of inequalities that describes the region.1.2. 3. 4.
(a) Graph the systems of inequalities representing the consumer surplus and producer surplus for the supply and demand equations (b) Find the consumer surplus and producer surplus. Demand ................................................... Supply 1. p = 50 − 0.5x
1. A person plans to invest up to $20,000 in two different interest-bearing accounts. Each account must contain at least $5000. The amount in one account is to be at least twice the amount in the other account. Write and graph a system of inequalities that describes the various amounts that can be
A dietician prescribes a special dietary plan using two different foods. Each ounce of food X contains 180 milligrams of calcium, 6 milligrams of iron, and 220milligrams of magnesium. Each ounce of food Y contains 100milligrams of calcium, 1milligram of iron, and 40milligrams of magnesium.
One formula for a person's maximum heart rate is 220 − x, where x is the person's age in years for 20 ≤ x ≤ 70. The American Heart Association recommends that when a person exercises, the person should strive for a heart rate that is at least 50% of the maximum and at most 85% of the maximum.
A physical fitness facility is constructing an indoor running track with space for exercise equipment inside the track (see figure). The track must be at least 125 meters long, and the exercise space must have an area of at least 500square meters.(a) Write and graph a system of inequalities that
Determine whether the statement is true or false. Justify your answer.1. The area of the figure described by the systemis 99 square units. 2. The graph shows the solution of the system 3. After graphing the boundary line of the inequality x + y
1. The graph of the solution of the inequality x + 2y (a) x + 2y ‰¤ 6(b) x + 2y > 62. Match the system of inequalities with the graph of its solution. [The graphs are labeled (i), (ii), (iii), and (iv).] (i) (ii) (iii) (iv) (a) (b) (c) (d)
Two concentric circles have radii x and y, where y > x. The area between the circles is at least 10 square units. (a) Write a system of inequalities that describes the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line y = x in the
Sketch the region corresponding to the system of constraints. Then find the minimum and maximum values of the objective function (if possible) and the points where they occur, subject to the constraints. 1. Objective function: z = 3x + 2y Constraints: x ≥ 0 y ≥ 0 3x + 2y ≤ 24 4x + y ≥
Use a graphing utility to graph the region corresponding to the system of constraints. Then find the minimum and maximum values of the objective function and the points where they occur, subject to the constraints. 1. Objective function: z = 3x + y Constrains: x ≥ 0 y ≥ 0 x + 4y ≤ 60 3x + 2y
Find the minimum and maximum values of the objective function and the points where they occur, subject to the constraints x ≥ 0, y ≥ 0, x + 4y ≤ 20, x + y ≤ 18, and 2x + 2y ≤ 21. 1. z = x + 5y 2. z = 2x + 4y
Find the minimum and maximum values (if possible) of the objective function and the points where they occur, subject to the constraints x ≥ 0, 3x + y ≥ 15, -x + 4y ≥ 8, and -2x + y ≥ -19. 1. z = x + 2y 2. z = 5x + 3y 3. z = x − y
The linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and the points where they occur. 1. Objective function: z = 2.5x +
A merchant plans to sell two models of MP3 players at prices of $225 and $250. The $225 model yields a profit of $30 per unit and the $250 model yields a profit of $31 per unit. The merchant estimates that the total monthly demand will not exceed 275 units. The merchant does not want to invest more
A manufacturer produces two models of elliptical cross-training exercise machines. The times for assembling, finishing, and packaging model X are 3 hours, 3 hours, and 0.8 hour, respectively. The times for model Y are 4 hours, 2.5 hours, and 0.4 hour. The total times available for
A public aquarium is adding coral nutrients to a large reef tank. The minimum amounts of nutrients A, B, and C that need to be added to the tank are 30 units, 16 units, and 24 units, respectively. Information about each bottle of brand X and brand Y additives is shown below. How many bottles of
A manufacturer has two different factories that produce three grades of steel: structural steel, rail steel, and pipe steel. They must produce 32 tons of structural, 26 tons of rail, and 30 tons of pipe steel to fill an order. The table shows the number of employees at each factory and the amounts
An accounting firm has 780 hours of staff time and 272 hours of reviewing time available each week. The firm charges $1600 for an audit and $250 for a tax return. Each audit requires 60hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of
The accounting firm in Exercise 41 lowers its charge for an audit to $1400. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue?
A fruit grower raises crops A and B. The yield is 300 bushels per acre for crop A and 500 bushels per acre for crop B. Research and available resources indicate the following constraints.The fruit grower has 150 acres of land available. It takes 1 day to trim the trees on an acre of crop A and 2
In Exercise 43, the profit is $185 per acre for crop A and $245 per acre for crop B. What is the optimal profit?
A company budgets a maximum of $1,000,000 for national advertising of an allergy medication. Each TV ad costs $100,000 and each one-page newspaper ad costs $20,000. Each TV ad is expected to be viewed by 20 million viewers, and each newspaper ad is expected to be seen by 5million readers. The
An investor has up to $450,000 to invest in two types of investments. Type A pays 6% annually and type B pays 10% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at
Determine whether the statement is true or false. Justify your answer. 1. If an objective function has a maximum value at the vertices (4, 7) and (8, 3), then it also has a maximum value at the points (4.5, 6.5) and (7.8, 3.2). 2. If an objective function has a minimum value at the vertex (20, 0),
Using the constraint region shown below, determine which of the following objective functions has (a) a maximum at vertex A, (b) a maximum at vertex B, (c) a maximum at vertex C, and (d) a minimum at vertex C.(i) z = 2x + y(ii) z = 2x ˆ’ y(iii) z = ˆ’x + 2y $y 5. A(0, 4) \B(4, 3) 3- 2- C(5, 0) 1+ х -i 12 3 4$
A linear programming problem has an objective function z = 3x + 5y and an infinite number of optimal solutions that lie on the line segment connecting two points. What is the slope between the points?
Use the graph of the region corresponding to the system of constraints to find the minimum and maximum values of the objective function subject to the constraints. Identify the points where the optimal values occur.1. Objective function:z = 4x + 3yConstraints:x ‰¥ 0y ‰¥ 0x + y
Solve the system by the method of substitution.1.2. 3.
1. Explain what is meant by an inconsistent system of linear equations. 2. How can you tell graphically that a system of linear equations in two variables has no solution? Give an example.
Solve the system graphically.1.2. 3. 4.
Use a graphing utility to solve the systems of equations. Round your solution(s) to two decimal places1.2. 3. 4.
1. Body Mass Index (BMI) is a measure of body fat based on height and weight. The 85th percentile BMI for females, ages 9 to 20, increases more slowly than that for males of the same age range. Models that represent the 85th percentile BMI for males and females, ages 9 to 20, arewhere B is the BMI
The perimeter of a rectangle is 68 feet and its width is 8/9 times its length. Use a system of equations to find the dimensions of the rectangle.
The perimeter of a rectangle is 40inches. The area of the rectangle is 96 square inches. Use a system of equations to find the dimensions of the rectangle.
Solve the system by the method of elimination and check any solutions algebraically.1.2. 3.
Match the system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent. [The graphs are labeled (a), (b), (c), and (d).]a.b. c. d. 1. 2. 3. 4.
Find the equilibrium point of the demand and supply equations. Demand...............................Supply 1. p = 43 − 0.0002x........ ..........p = 22 + 0.00001x 2. p = 120 − 0.0001x................ p = 45 + 0.0002x
Use back-substitution to solve the system of linear equations.1.2.
Solve the system of linear equations and check any solutions algebraically.1.2. 3.
Solve the system of linear equations and check any solutions algebraically.1.2. $(2r + 5y – 19z = 34 19г %3D \ 3х + 8y — 31z %3D 54$
find the equation of the parabola y = ax2 + bx + c that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabolas.1.2.
Find the equation of the circle x2 + y2 + Dx + Ey + F = 0 that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle.
A mixture of 6 gallons of chemical A, 8gallons of chemical B, and 13 gallons of chemical C is required to kill a destructive crop insect. Commercial spray X contains one, two, and two parts, respectively, of these chemicals. Commercial spray Y contains only chemical C. Commercial spray Z
The Old Course at St Andrews Links in St Andrews, Scotland, is one of the oldest golf courses in the world. It is an 18-hole course that consists of par-3 holes, par-4 holes, and par-5 holes. There are seven times as many par-4 holes as par-5 holes, and the sum of the numbers of par-3 and par-5
An inheritance of $40,000 is divided among three investments yielding $3500 in interest per year. The interest rates for the three investments are 7%, 9%, and 11% simple interest. Find the amount placed in each investment when the second and third amounts are $3000 and $5000 less than the first,
An amount of $46,000 is divided among three investments yielding $3020 in interest per year. The interest rates for the three investments are 5%, 7%, and 8% simple interest. Find the amount placed in each investment when the second and third amounts are $2000 and $3000 less than the first,
An object moving vertically is at the given heights at the specified times. Find the position equation s = 1/2at2 + v0t + s0 for the object. At t = 1 second, s = 134 feet At t = 2 seconds, s = 86 feet At t = 3 seconds, s = 6 feet
Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 1. 3 / x2 + 20x 2. x - 8 / x2 - 3x - 28 3. 3x - 4 / x3 - 5x2 4. x - 2 / x(x2 + 2)2
Write the partial fraction decomposition of the rational expression. Check your result algebraically. 1. 4 - x / x2 + 6x 8 2. - x / x2 + 3x + 2 3. x2 / x2 + 2x - 15
Sketch the graph of the inequality. 1. y ≤ 5 2. x < - 3 3. y ≤ 5 - 2x 4. 3y - x ≥ 7 5. (x - 1)2 + (y - 3)2 < 16
Sketch the graph of the solution set of the system of inequalities. Label the vertices of the region.
Sketch the graph of the solution set of the system of inequalities.1.2.
Write a system of inequalities to describe the region of a rectangle with vertices at (3, 1), (7, 1), (7, 10), and (3, 10).
Write a system of inequalities that describes the triangular region with vertices (0, 5), (5, 0), and (0, 0).
(a) Graph the systems of inequalities representing the consumer surplus and producer surplus for the supply and demand equations. (b) Find the consumer surplus and producer surplus. Demand Supply p = 160 − 0.0001x p = 70 + 0.0002x
A warehouse operator has 24,000square feet of floor space in which to store two products. Each unit of product I requires 20 square feet of floor space and costs $12 per day to store. Each unit of product II requires 30 square feet of floor space and costs $8 per day to store. The total storage
A dietician prescribes a special dietary plan using two different foods. Each ounce of food X contains 200 milligrams of calcium, 3 milligrams of iron, and 100 milligrams of magnesium. Each ounce of food Y contains 150 milligrams of calcium, 2milligrams of iron, and 80milligrams of
Sketch the region corresponding to the system of constraints. Then find the minimum and maximum values of the objective function (if possible) and the points where they occur, subject to the constraints. 1. Objective function: z = 3x + 4y Constrains: x ≥ 0 t ≥ 0 2x + 5y
A student is working part time as a hairdresser to pay college expenses. The student may work no more than 24 hours per week. Haircuts cost $25 and require an average of 20 minutes, and permanents cost $70 and require an average of 1 hour and 10 minutes. How many haircuts andor permanents will
A manufacturer produces two models of bicycles. The table shows the times (in hours) required for assembling, painting, and packaging each model.The total times available for assembling, painting, and packaging are 4000 hours, 4800 hours, and 1500hours, respectively. The profits per unit are $45
Determine whether the statement is true or false. Justify your answer.The systemrepresents a region in the shape of an isosceles trapezoid.
Write a system of linear equations that has the ordered pair as a solution. (There are many correct answers.) 1. (−8, 10) 2. (5, −4) 3. (4/3, 3)
Write a system of linear equations that has the ordered triple as a solution. (There are many correct answers.) 1. (4, −1, 3) 2. (−3, 5, 6) 3. (5, 3/2, 2)
A matrix is ________ when the number of rows equals the number of columns. 2. For a square matrix, the entries a11, a22, a33, . . . are the ________ ________ entries. 3. A matrix derived from a system of linear equations (each written in standard form with the constant term on the right) is the
Determine whether the matrix below is in row-echelon form, reduced row-echelon form, or neither when it satisfies the given conditions.(a) b = 0, c = 0 (b) b ‰ 0, c = 0 (c) b = 0, c ‰ 0 (d) b ‰ 0, c ‰ 0
Write the augmented matrix for the system of linear equations.1.2. 3. 4. 5. 6.
Write the system of linear equations represented by the augmented matrix. (Use variables x, y, z, and w, if applicable.)1.2. 3. 4. 5. 6. 7.
Identify the elementary row operation(s) performed to obtain the new row-equivalent matrix.1.Original Matrix New Row-Equivalent Matrix2. Original Matrix New Row-Equivalent Matrix 3. Original Matrix New Row-Equivalent Matrix 4. Original Matrix New Row-Equivalent Matrix
Fill in the blank(s) using elementary row operations to form a row-equivalent matrix.1.2. 3. 4.
(a) Perform the row operations to solve the augmented matrix,(b) Write and solve the system of linear equations (in variables x, y, and z, if applicable) represented by the augmented matrix, and(c) Compare the two solution methods. Which do you prefer?1.(i) Add R2 to R1. (ii) Add - 2 times R1 to
Determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.1.2. 3. 4.
Write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)1.2. 3. 4.
Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.1.2. 3. 4. 5. 6.
Write the system of linear equations represented by the augmented matrix. Then use back substitution to solve the system. (Use variables x, y, and z, if applicable.)1.2. 3. 4.
Use matrices to solve the system of linear equations, if possible. Use Gaussian elimination with back-substitution.1.2. 3.
An augmented matrix that represents a system of linear equations (in variables x, y, and z, if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix.1.2.
Determine the dimension of the matrix.1. [7 0]2. [5 -3 8 7]3.4. 5. 6. 7. 8.
Use matrices to solve the system of linear equations, if possible. Use Gauss-Jordan elimination.1.2.
Use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of linear equations in reduced row-echelon form. Then solve the system.1.2.
1. Describe the error. The matrixhas four rows and one column, so the dimension of the matrix is 1 Ã— 4. 2. Describe the error. The matrix is in reduced row-echelon form.
Use a system of linear equations to find the quadratic function f (x) = ax2 + bx + c that satisfies the given conditions. Solve the system using matrices. 1. f (1) = 1, f (2) = −1, f (3) = −5 2. f (1) = 2, f (2) = 9, f (3) = 20
From 2005 through 2016, the numbers of new cases of a waterborne disease in a small city increased in a pattern that was approximately linear (see figure). Find the least squares regression line y = at + b for the data shown in the figure by solving the system below using matrices. Let t represent
A natural history museum borrows $2,000,000 at simple annual interest to purchase new exhibits. Some of the money is borrowed at 7%, some at 8.5%, and some at 9.5%. Use a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $169,750 and
A city zoo borrows $2,000,000 at simple annual interest to construct a breeding facility. Some of the money is borrowed at 8%, some at 9%, and some at 12%. Use a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $186,000 and the amount
Mathematical Modeling A video of the path of a ball thrown by a baseball player was analyzed with a grid covering the TV screen. The video was paused three times, and the position of the ball was measured each time. The coordinates obtained are shown in the table. (x and y are measured in feet.)(a)
Determine whether the statement is true or false. Justify your answer.1.is a 4 Ã— 2 matrix. 2. The method of Gaussian elimination reduces a matrix until a reduced row-echelon form is obtained. 3. What is the relationship between the three elementary row operations performed on an
The columns of matrix T show the coordinates of the vertices of a triangle. Matrix A is a transformation matrix.(a) Find AT and AAT. Then sketch the original triangle and the two images of the triangle. What transformation does A represent? (b) Given the triangle determined by AAT, describe the
Verify the following equation.
1. Verify the following equation.2. Find a 4 Ã— 4 matrix whose determinant is equal to ax3 + bx2 + cx + d. (Hint: Use the equation in Exercise 12 as a model.)
Let A be an n × n matrix each of whose rows sum to zero. Find (A(.
The table shows the masses (in atomic mass units) of three compounds. Use a linear system and Cramer's Rule to find the atomic masses of sulfur(S), nitrogen (N), and fluorine (F).
A walkway lighting package includes a transformer, a certain length of wire, and a certain number of lights on the wire. The price of each lighting package depends on the length of wire and the number of lights on the wire. Use the information below to find the cost of a transformer, the cost per
Use the inverse of A to decode the cryptogram.23 13 ˆ’34 31 ˆ’34 63 25 ˆ’17 61 24 14 ˆ’37 41 ˆ’17 ˆ’8 20 ˆ’29 40 38 ˆ’56 116 13 ˆ’11 1 22 ˆ’3 ˆ’6 41 ˆ’53
A code breaker intercepts the encoded message below.45 ˆ’35 38 ˆ’30 18 ˆ’18 35 ˆ’30 81 ˆ’6042 ˆ’28 75 ˆ’55 2 ˆ’2 22 ˆ’21 15 ˆ’10(a) You know that Where A-1 is the inverse of the encoding matrix A.
The matrices show the male and female populations in the United States in 2011 and 2014. The male and female populations are separated into three age groups.(a) The total population in 2011 was 311,721,632 and the total population in 2014 was 318,857,056. Rewrite the matrices to give the
Consider matrices of the form(a) Write a 2 Ã— 2 matrix and a 3 Ã— 3 matrix in the form of A. (b) Use a graphing utility to raise each of the matrices to higher powers. Describe the result. (c) Use the result of part (b) to make a conjecture about powers of A when A is a 4
A square matrix is idempotent when A2 = A. Determine whether each matrix is idempotent.(a)(b) (c) (d) (e) (f)

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