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College Algebra Graphs and Models 5th edition Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna - Solutions

State the conditions under which A-1 exists. Then find a formula for A-1.a. A = [ x ]b.
Use the following matrix for Exercises:a. Find M12 and M44. b. Find M41 and M33.
Determine whether the function is one-to-one, and if it is, find a formula for f -1(x). a. f(x) = 3x + 2 b. f(x) = x2 - 4
Simplify. Write answers in the form a + bi, where a and b are real numbers. a. (3 - 4i) - (-2 - i) b. (5 + 2i) + (1 - 4i) c. (1 - 2i) (6 + 2i)
Use the following matrix for Exercises:a. Find M11, M32, and M22. b. Find M13, M31, and M23. c. Find A11, A32, and A22.
In Exercises, match the inequality with one of the graphs (a)-(h), which follow.a. y > x b. y c. y ‰¤ x - 3 d. y ‰¥ x + 5 e. 2x + y
In Exercises, match the system of inequalities with one of the graphs (a)-(f), which follow.1. y > x + 1, y ‰¤ 2 - x 2. y y ‰¥ 4 - x 3. 2x + y 4x + 2y > 12 4. x ‰¤ 5, y ‰¥ 1 5. x + y ‰¤ 4, x - y ‰¥ -3, x ‰¥ 0, y ‰¥
Find a system of inequalities with the given graph.a.b.
Graph the system of inequalities. Then find the coordinates of the vertices. a. y ≤ x, y ≥ 3 - x b. y ≤ x, y ≥ 5 - x c. y ≥ x, y ≤ 4 - x
Find the maximum value and the minimum value of the function and the values of x and y for which they occur. a. P = 17x - 3y + 60, subject to 6x + 8y ≤ 48, 0 ≤ y ≤ 4, 0 ≤ x ≤ 7 b. Q = 28x - 4y + 72, subject to 5x + 4y ≥ 20, 0 ≤ y ≤ 4, 0 ≤ x ≤ 3
Golden Harvest Foods makes jumbo biscuits and regular biscuits. The oven can cook at most 200 biscuits per day. Each jumbo biscuit requires 2 oz of flour, each regular biscuit requires 1 oz of flour, and there is 300 oz of flour available. The income from each jumbo biscuit is $0.80 and from each
Omar owns a pickup truck and a moped. He can afford 12 gal of gasoline to be split between the truck and the moped. Omar's truck gets 20 mpg and, with the fuel currently in the tank, can hold at most an additional 10 gal of gas. His moped gets 100 mpg and can hold at most 3 gal of gas. How many
Norris Mill can convert logs into lumber and plywood. In a given week, the mill can turn out 400 units of production, of which 100 units of lumber and 150 units of plywood are required by regular customers. The profit is $20 per unit of lumber and $30 per unit of plywood. How many units of each
Sunnydale Farm includes 240 acres of cropland. The farm owner wishes to plant this acreage in corn and oats. The profit per acre in corn production is $40 and in oats is $30. A total of 320 hr of labor is available. Each acre of corn requires 2 hr of labor, whereas each acre of oats requires 1 hr
An animal feed to be mixed from soybean meal and oats must contain at least 120 lb of protein, 24 lb of fat, and 10 lb of mineral ash. Each 100-lb sack of soybean meal costs $15 and contains 50 lb of protein, 8 lb of fat, and 5 lb of mineral ash. Each 100-lb sack of oats costs $5 and contains 15 lb
Suppose that in the preceding problem the oats were replaced by alfalfa, which costs $8 per 100 lb and contains 20 lb of protein, 6 lb of fat, and 8 lb of mineral ash. How much of each is now required in order to minimize the cost?
Clayton is planning to invest up to $40,000 in corporate and municipal bonds. The least he is allowed to invest in corporate bonds is $6000, and he does not want to invest more than $22,000 in corporate bonds. He also does not want to invest more than $30,000 in municipal bonds. The interest is 8%
Margaret is planning to invest up to $22,000 in certificates of deposit at City Bank and People's Bank. She wants to invest at least $2000 but no more than $14,000 at City Bank. People's Bank does not insure more than a $15,000 investment, so she will invest no more than that in People's Bank. The
An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide transportation for a minimum of 2000 first-class, 1500 tourist-class, and 2400 economy-class passengers. For a certain trip, airplane P1 costs $12 thousand to operate and can accommodate 40 first-class,
Suppose that in the preceding problem a new airplane P3 becomes available, having an operating cost for the same trip of $15 thousand and accommodating 40 firstclass, 40 tourist-class, and 80 economy-class passengers. If airplane P1 were replaced by airplane P3, how many of P2 and P3 should be used
It takes Just Sew 2 hr of cutting and 4 hr of sewing to make a knit suit. It takes 4 hr of cutting and 2 hr of sewing to make a worsted suit. At most 20 hr per day are available for cutting and at most 16 hr per day are available for sewing. The profit is $34 on a knit suit and $31 on a worsted
Cambridge Metal Works manufactures two sizes of gears. The smaller gear requires 4 hr of machining and 1 hr of polishing and yields a profit of $25. The larger gear requires 1 hr of machining and 1 hr of polishing and yields a profit of $10. The firm has available at most 24 hr per day for
Suppose that it takes 12 units of carbohydrates and 6 units of protein to satisfy Jacob's minimum weekly requirements. A particular type of meat contains 2 units of carbohydrates and 2 units of protein per pound. A particular cheese contains 3 units of carbohydrates and 1 unit of protein per pound.
The Spring Hill school board is analyzing education costs for Hill Top School. It wants to hire teachers and teacher's aides to make up a faculty that satisfies its needs at minimum cost. The average annual salary for a teacher is $46,000 and for a teacher's aide is $21,000. The school building can
A certain area of forest is populated by two species of animal, which scientists refer to as A and B for simplicity. The forest supplies two kinds of food, referred to as F1 and F2. For one year, each member of species A requires 1 unit of F1 and 0.5 unit of F2. Each member of species B requires
In Exercises 1-6, match the equation with one of the graphs (a)-(f), which follow.a.b. c. d. e. f. 1. x2 = 8 2. y2 = = -10x 3. (y-2)2 = -3(x +4) 4. (x+1)2 = 5(x - y)
Find an equation of a parabola satisfying the given conditions. a. Vertex 10, 02, focus 1-3, 02 b. Vertex 10, 02, focus 10, 102 c. Focus 17, 02, directrix x = -7
Find the vertex, the focus, and the directrix. Then draw the graph. a. (x + 2)2 = -61y - 12 b. (y - 3)2 = -201x + 22 c. x2 + 2x + 2y + 7 = 0
An engineer designs a satellite dish with a parabolic cross section. The dish is 15 ft wide at the opening, and the focus is placed 4 ft from the vertex.a) Position a coordinate system with the origin at the vertex and the x-axis on the parabola's axis of symmetry and find an equation of the
A heavy-duty flashlight mirror has a parabolic cross section with diameter 6 in. and depth 1 in.a) Position a coordinate system with the origin at the vertex and the x-axis on the parabola's axis of symmetry and find an equation of the parabola. b) How far from the vertex should the bulb be
Information Unlimited designed and sells the Ultrasonic Receiver, which detects sounds unable to be heard by the human ear. The HT90P can detect mechanical and electrical sounds such as leaking gases, air, corona, and motor friction noises. It can also be used to hear bats, insects, and even
A spotlight has a parabolic cross section that is 4 ft wide at the opening and 1.5 ft deep at the vertex. How far from the vertex is the focus?
Consider the following linear equations. Without graphing them, answer the questions below.a. y = 2xb. y = 1/3 x + 5c. y = -3x -2d. y = -0.9x + 7e. y = -5x + 3f. y = x + 4g. 8x - 4y = 7h. 3x + 6y = 21. Which has/have x-intercept (2/3, 0)?2. Which has/have y-intercept (0, 7)?3. Which slant up from
Find an equation of the parabola with a vertical axis of symmetry and vertex (-1, 2) and containing the point (-3, 1).
Find an equation of a parabola with a horizontal axis of symmetry and vertex (-2, 1) and containing the point (-3, 5).
The cables of a 200-ft portion of the roadbed of a suspension bridge are positioned as shown below. Vertical cables are to be spaced every 20 ft along this portion of the roadbed. Calculate the lengths of these vertical cables.
Find the vertex, the focus, and the directrix. Then draw the graph. a. x2 = 20y b. y2 = -6x c. x2 = 16y
In Exercises 1-6, match the equation with one of the graphs (a)-(f), which follow.a.b. c. d. e. 1. x2 + y2 = 5 2. y2 = 20 - x2 3. x2 + y2 - 6x + 2y = 6 4. x2 + y2 + 10x - 12y = 3
In Exercises 19-22, match the equation with one of the graphs (a)-(d), which follow.a.b. c. d. 1. 16x2 + 4y2 = 64 2. 4x2 + 5y2 = 20 3. x2 + 9y2 - 6x + 90y = -225 4. 9x2 + 4y2 + 18x - 16y = 11
Find the vertices and the foci of the ellipse with the given equation. Then draw the graph. a. x2 / 4 + y2 / 1 = 1 b. x2 / 25 + y2 / 36 = 1 c. 16x2 + 9y2 = 144
Find an equation of an ellipse satisfying the given conditions. a. Vertices: (-7, 0) and (7, 0); foci: (-3, 0) and (3, 0) b. Vertices: (0, -6) and (0, 6); foci: (0, -4) and (0, 4) c. Vertices: (0, -8) and (0, 8); length of minor axis: 10
Find the center, the vertices, and the foci of the ellipse. Then draw the graph. a. (x - 1)2 / 9 + (y - 2)2/4 = 1 b. (x - 1)2/1 + (y - 2)2/4 = 1
Observe the shapes of the ellipses in Examples 2 and 4. Which ellipse has the smaller eccentricity? Confirm your answer by computing the eccentricity of each ellipse. The eccentricity of an ellipse is defined as e = c / a. For an ellipse, 0 < c < a, so 0 < e < 1, when e is close to 0, an ellipse
Find an equation of an ellipse with vertices (0, -4) and (0, 4) and e = 1 / 4. The eccentricity of an ellipse is defined as e = c / a. For an ellipse, 0 < c < a, so 0 < e < 1, when e is close to 0, an ellipse appears to be nearly circular. When e is close to 1, an ellipse is very flat.
Find an equation of an ellipse with vertices (-3, 0) and (3, 0) and e = 7 / 10. The eccentricity of an ellipse is defined as e = c / a. For an ellipse, 0 < c < a, so 0 < e < 1, when e is close to 0, an ellipse appears to be nearly circular. When e is close to 1, an ellipse is very flat.
The bridge support shown in the figure below is the top half of an ellipse. Assuming that a coordinate system is superimposed on the drawing in such a way that point Q, the center of the ellipse, is at the origin, find an equation of the ellipse.The eccentricity of an ellipse is defined as e = c /
In Washington, D.C., there is a large grassy area south of the White House known as the Ellipse. It is actually an ellipse with major axis of length 1048 ft and minor axis of length 898 ft. Assuming that a coordinate system is superimposed on the area in such a way that the center is at the origin
The maximum distance of the earth from the sun is 9.3 × 107 mi. The minimum distance is 9.1 × 107 mi. The sun is at one focus of the elliptical orbit. Find the distance from the sun to the other focus. The eccentricity of an ellipse is defined as e = c / a. For an ellipse, 0 < c < a, so 0 < e <
A carpenter is cutting a 3-ft by 4-ft elliptical sign from a 3-ft by 4-ft piece of plywood. The ellipse will be drawn using a string attached to the board at the foci of the ellipse.a) How far from the ends of the board should the string be attached? b) How long should the string be? The
Find an equation of an ellipse satisfying the given conditions. a. Vertices: (3, -4), (3, 6); endpoints of minor axis: (1, 1), (5, 1) b. Vertices: (-1, -1), (-1, 5); endpoints of minor axis: (-3, 2), (1, 2)
A bridge with a semielliptical arch spans a river as shown here. What is the clearance 6 ft from the riverbank?
Find the center and the radius of the circle with the given equation. Then draw the graph. 1. x2 + y2 - 14x + 4y = 11 2. x2 + y2 + 2x - 6y = -6 3. x2 + y2 + 6x - 2y = 6
In Exercises 1-6, match the equation with one of the graphs (a)-(f), which follow.a.b. c. d. e. f. 1. x2 / 25 - y2 / 9 = 1 2. y2 / 4 - x2 / 36 = 1 3. (y - 1)2 / 16 - (x + 3)2 /1 = 1 4. (x + 4)2 / 100 - (y - 2)2 / 81 = 1
Find the center, the vertices, the foci, and the asymptotes. Then draw the graph. a. x2/4 - y2/4 = 1 b. x2/1 - y2/9 = 1
Find the center, the vertices, the foci, and the asymptotes of the hyperbola. Then draw the graph. a. x2 - y2 - 2x - 4y - 4 = 0 b. 4x2 - y2 + 8x - 4y - 4 = 0
Observe the shapes of the hyperbolas in Examples 2 and 3. Which hyperbola has the larger eccentricity? Confirm your answer by computing the eccentricity of each hyperbola. The eccentricity of a hyperbola is defined as e = c > a. For a hyperbola, c > a > 0, so e > 1 when e is close to 1, a hyperbola
Find an equation of a hyperbola with vertices (3, 7) and (-3, 7) and e = 5/3. The eccentricity of a hyperbola is defined as e = c > a. For a hyperbola, c > a > 0, so e > 1 when e is close to 1, a hyperbola appears to be very narrow as the eccentricity increases, the hyperbola becomes "wider."
Find an equation of a hyperbola with vertices (-1, 3) and (-1, 7) and e = 4. The eccentricity of a hyperbola is defined as e = c > a. For a hyperbola, c > a > 0, so e > 1 when e is close to 1, a hyperbola appears to be very narrow as the eccentricity increases, the hyperbola becomes "wider."
A cross section of a nuclear cooling tower is a hyperbola with equationx2 / 902 - y2 / 1302 = 1.The tower is 450 ft tall, and the distance from the top of the tower to the center of the hyperbola is half the distance from the base of the tower to the center of the hyperbola. Find the diameter of
Certain telescopes contain both a parabolic mirror and a hyperbolic mirror. In the telescope shown in the figure, the parabola and the hyperbola share focus F1, which is 14 m above the vertex of the parabola. The hyperbola's second focus F2 is 2 m above the parabola's vertex. The vertex of the
In Exercises given the function: a) Determine whether it is one-to-one. b) If it is one-to-one, find a formula for the inverse. 1. f(x) = 2x - 3 2. f(x) = x3 + 2
Find an equation of a hyperbola satisfying the given conditions. a. Vertices: (3, -8) and (3, -2); asymptotes: y = 3x - 14, y = -3x + 4 b. Vertices: (-9, 4) and (-5, 4); asymptotes: y = 3x + 25, y = -3x - 17
Use a graphing calculator to find the center, the vertices, and the asymptotes. a. 5x2 - 3.5y2 + 14.6x - 6.7y + 3.4 = 0 b. x2 - y2 - 2.046x - 4.088y - 4.228 = 0
Two radio transmitters positioned 300 mi apart along the shore send simultaneous signals to a ship that is 200 mi offshore, sailing parallel to the shoreline. The signal from transmitter S reaches the ship 200 microseconds later than the signal from transmitter T. The signals travel at a speed of
Find an equation of a hyperbola satisfying the given conditions. a. Vertices: (0, 3) and (0, -3); foci: (0, 5) and (0, -5) b. Vertices: (1, 0) and (-1, 0); foci: (2, 0) and (-2, 0) c. Asymptotes: y = 3/2 x, y = - 3/2 x; one vertex: (2, 0)
In Exercises 1-6, match the system of equations with one of the graphs (a)-(f), which follow.1. x2 + y2 = 16, x + y = 3 2. 16x2 + 9y2 = 144, x - y = 4 3. y = x2 - 4x - 2, 2y - x = 1 4. 4x2 - 9y2 = 36, x2 + y2 = 25 5. y = x2 - 3, x2 + 4y2 = 16 6. y2 - 2y = x + 3, xy = 4
Solve using a graphing calculator. Find all real solutions. a. y - ln x = 2, y = x2 b. y = ln (x + 4), x2 + y2 = 6 c. y = ex, x - y = -2 d. y - e-x = 1, y = 2x + 5
Frank's Frame Shop is building a frame for a rectangular oil painting with a perimeter of 28 cm and a diagonal of 10 cm. Find the dimensions of the painting.
Peden's Advertising is building a rectangular sign with an area of 2 yd2 and a perimeter of 6 yd. Find the dimensions of the sign.
Marcia Graham, owner of Graham's Graphics, is designing an advertising brochure for the Art League's spring show. Each page of the brochure is rectangular with an area of 20 in2 and a perimeter of 18 in. Find the dimensions of the brochure.
Green Leaf Landscaping is planting a rectangular wildflower garden with a perimeter of 6 m and a diagonal of 15 m. Find the dimensions of the garden.
It will take 210 yd of fencing to enclose a rectangular dog pen. The area of the pen is 2250 yd2. What are the dimensions of the pen?
Ted Hansen of Hansen Woodworking Designs has been commissioned to make a rectangular tabletop with an area of √2 m2 and a diagonal of √3 m for the Decorators' Show House. Find the dimensions of the tabletop.
A rectangular banner with an area of ˆš3 m2 is being designed to advertise an exhibit at the Davis Gallery. The length of a diagonal is 2 m. Find the dimensions of the banner.
Jenna made an investment for 1 year that earned $7.50 simple interest. If the principal had been $25 more and the interest rate 1% less, the interest would have been the same. Find the principal and the interest rate.
The Burton Seed Company has two square test plots. The sum of their areas is 832 ft2, and the difference of their areas is 320 ft2. Find the length of a side of each plot
The diagonal of the floor of a rectangular office cubicle is 1 ft longer than the length of the cubicle and 3 ft longer than twice the width. Find the dimensions of the cubicle.
Graph the system of inequalities. Then find the coordinates of the points of intersection of the graphs of the related equations. a. x2 + y2 ≤ 16, y < x b. x2 + y2 ≤ 10, y > x
Find an equation of the circle that passes through the points (2, 4) and (3, 3) and whose center is on the line 3x - y = 3.
Find an equation of the circle that passes through the points (2, 3), (4, 5), and (0, -3).
Find an equation of an ellipse centered at the origin that passes through the points (1, √3 / 2) and (√3, 1/2).
Find an equation of a hyperbola of the type x2 / b2 - y2 / a2 = 1 That passes through the points (-3, -3√5/2) and (-3 / 2, 0).
Find two numbers whose product is 2 and the sum of whose reciprocals is 33/8 .
Four squares with sides 5 in. long are cut from the corners of a rectangular metal sheet that has an area of 340 in2. The edges are bent up to form an open box with a volume of 350 in3. Find the dimensions of the box.
Find the vertex, the focus, and the directrix of the parabola. Then draw the graph. a. y2 = 12x b. x2 - 6x - 4y = -17
Find the equation of a parabola satisfying the given conditions. a. Focus: (0, 3); directrix: y = 1 b. Focus: (-4, 6); directrix: x = 2
Find the center and the radius of the circle. Then draw the graph. a. x2 + y2 + 4x - 8y = 5 b. x2 + y2 - 6x + 2y - 6 = 0
Find the vertices and the foci of the ellipse. Then draw the graph a. x2 / 1 + y2 / 9 = 1 b. 2x2 + 3y2 = 12
Write an equation of the ellipse satisfying the given conditions. a. Vertices: (-5, 0), (5, 0); foci: (-2, 0), (2, 0) b. Vertices: (0, -3), (0, 3); length of minor axis: 4 c. Foci: (-3, 0), (3, 0); length of major axis: 8
In Exercises 5-12, match the equation with one of the graphs (a)-(h), which follow. [a.b. c. d. e. f. g. h. 1. x2 = -4y 2. 1y + 222 = 41x - 22 3. 16x2 + 9y2 = 144 4. x2 + y2 = 16
Find the vertex, the focus, and the directrix of the parabola given by x2 + 10x + 2y + 9 = 0.
Find the center, the vertices, and the foci of the ellipse given by 16x2 + 25y2 - 64x + 50y - 311 = 0 Then draw the graph.
Find an equation of the ellipse having vertices (0, -4) and (0, 4) with minor axis of length 6.
Find the center, the vertices, the foci, and the asymptotes of the hyperbola given by x2 - 2y2 + 4x + y - 1/8 = 0.
A spotlight has a parabolic cross section that is 2 ft wide at the opening and 1.5 ft deep at the vertex. How far from the vertex is the focus?
The sum of two numbers is 11, and the sum of their squares is 65. Find the numbers.
A rectangle has a perimeter of 38 m and an area of 84 m2. What are the dimensions of the rectangle?
Find two positive integers whose sum is 12 and the sum of whose reciprocals is 38.
The perimeter of a square is 12 cm more than the perimeter of another square. The area of the first square exceeds the area of the other by 39 cm2. Find the perimeter of each square.
The sum of the areas of two circles is 130p ft2. The difference of the areas is 112p ft2. Find the radius of each circle.

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