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College Algebra With Modeling And Visualization 6th Edition Gary Rockswold - Solutions

Write the given equation in either the form (yk)² = a (x - h) or (xh)² = a(y-k).
Find an equation of a parabola that satisfies the given conditions.Focus (1, 2) and directrix y = 4
Find an equation of a parabola that satisfies the given conditions.Horizontal axis, vertex (-2,3), passing through (-4,0)
Find an equation of a parabola that satisfies the given conditions.Horizontal axis, vertex (-1, 2), passing through (2, 3)
Write the given equation in either the form (yk)² = a (x - h) or (xh)² = a(y-k).
Solve the system of equations. Give graphical support by making a sketch. 4x² + y² = 4 x² + 1² = 2
Solve the system of equations. Give graphical support by making a sketch. x² + y² = 4 (x - 1)² + y² = 4
Explain how the center, vertices, and asymptotes of a hyperbola are related to the fundamental rectangle.
Suppose that the coordinates of F are (0, 5.2) and the coordinates of F are (0, -5.2). If the coordinates of the vertex of the hyperbolic mirror are (0, 4.1), find the standard equation of a hyperbola whose upper branch coin- cides with the hyperbolic mirror.
Solve the system of equations. Give graphical support by making a sketch. x² + ² = 9 2x² + 3y² = 18
Shade the solution set to the system. (x - 1)² + (y + 1)² < 4 (x + 1)² + y² > 1
Shade the solution set to the system. 16 4 + 25 V 1
Shade the solution set to the system. + ≤1 4 x + y = 2
Shade the solution set to the system. x² + y² ≤ 4 x² + (y-2)² = 4
Shade the solution set to the system. 1 16 25 -x+y ≤4
Given the standard equation of a hyperbola, explain how to determine the length of the transverse axis. How can you determine whether the transverse axis is vertical or horizontal?
Shade the solution set to the system. x² + (y + 1)² ≤ 9 (x + 1)² + y² ≤ 9
Shade the solution set to the system. x² + y² ≤4 (x + 1)²-y≤0
Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. (x-1)²2 (y + 2)² + 25 16 ≤1
Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. 98 = 16+ 2x7
Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. (x + 3)² 4 (y-2)² 8 ≤1
The radio telescope shown in the figure has the shape of a parabolic dish with a diameter of 210 feet and a depth of 32 feet.(a) Determine an equation of the form y = ax2 with a > 0 describing a cross section of the dish. (b) The receiver is placed at the focus. How far from the vertex is the
Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. 9x² + y² ≤ 9
Shade the solution set to the system. 4x² +9y² ≤ 36 x-(y-2)² ≥ 0
A headlight is being constructed in the shape of a paraboloid with a depth of 4 inches and a diameter of 5 inches, as illustrated in the figure. Find the distance d that the bulb should be from the vertex in order to have the beam of light shine straight ahead. d 4 in.- 5 in. J
A solar heater is being designed to heat a pipe that will contain water, as illustrated in the figure. A cross section of the heater is described by the equation x = ky, where k is a constant and all units are in feet. If the pipe is to be placed 18 inches from the vertex of this cross section,
Use the dimensions of a television satellite dish in the shape of a paraboloid to calculate how far from the vertex the receiver should be located.Six-foot diameter, nine inches deep
A radio telescope is being designed in the shape of a parabolic dish with a diameter of 180 feet and a depth of 25 feet. (a) Determine an equation of the form x = ay2 with a > 0 describing a cross section of the dish. (b) The receiver is placed at the focus. How far from the vertex
Use the dimensions of a television satellite dish in the shape of a paraboloid to calculate how far from the vertex the receiver should be located.Nine-inch radius, two inches deep
A comet sometimes travels along a parabolic path as it passes the sun. In this case the sun is located at the focus of the parabola and the comet passes the sun once, rather than orbiting the sun. Suppose the path of a comet is given by y2 = 100x, where units are in millions of miles. (a) Find
Find an equation for the orbit of the planet. Graph its orbit and the location of the sun at a focus on the positive x-axis.Mercury: e = 0.206, a = 0.387
Find an equation for the orbit of the planet. Graph its orbit and the location of the sun at a focus on the positive x-axis.Mars: e = 0.093, a = 1.524
The source of a shock wave is placed at one focus of an ellipsoid with a major axis of 8 inches and a minor axis of 5 inches. Estimate, to the near- est thousandth of an inch, how far a kidney stone should be positioned from the source.
Explain how the distance between the focus and the vertex of a parabola affects the shape of the parabola.
Explain how to determine the direction that a parabola opens, given the focus and the directrix.
The maximum and minimum velocities in kilometers per second of a celestial body moving in an elliptical orbit can be calculated byIn these equations, a is half the length of the major axis of the orbit in kilometers, P is the orbital period in seconds, and e is the eccentricity of the orbit. (a)
The perimeter P of an ellipse can be approximated by(a) Approximate the distance in miles that Mercury travels in one orbit of the sun if a = 36.0, b = 35.2, and the units are in millions of miles. (b) If a planet has a circular orbit, does this formula give the exact perimeter? Explain. P =
The orbit of Explorer VII and the outline of Earth's surface are shown in the figure at the top of the next column. This orbit can be described by the equation x2/z2 + y2/b2 = 1, where a = 4464 and b = 4462. The surface of Earth can be described by (x - 164)2 + y2 = 39602. Find the maximum and
An elliptical arch under a bridge is con- structed so that it is 60 feet wide and has a maximum height of 25 feet, as illustrated in the figure. Find the height of the arch 15 feet from the center of the arch. 25 ft 60 ft
Halley's comet travels in an ellipti- cal orbit with a = 17.95 and b = 4.44 and passes by Earth roughly every 76 years. Note that each unit represents one astronomical unit, or 93 million miles. The comet most recently passed by Earth in February 1986.(a) Write an equation for this orbit, centered
A large room constructed in the shape of the upper half of an ellipsoid has a unique property. Any sound emanating from one focus is reflected directly toward the other focus. See the figure. If the foci are 100 feet apart and the maximum height of the ceiling is 40 feet, estimate the area of the
A patient's kidney stone is placed 12 units away from the source of the shock waves of a lithotripter. The lithotripter is based on an ellipse with a minor axis that measures 16 units. Find the equation of an ellipse that would satisfy this situation.
The perimeter of the Roman Colosseum is an ellipse with major axis 620 feet and minor axis 513 feet. Find the distance between the foci of this ellipse.
Earth has a nearly circular orbit with e ≈ 0.0167 and a = 93 million miles. Approximate the minimum and maximum distances between Earth and the sun.
Both Neptune and Pluto travel around the sun in elliptical orbits. For Neptune's orbit, a = 30.10, and for Pluto's orbit, a = 39.44, where the variable a represents their average distances from the sun in astronomical units. (One astronomical unit equals 93 million miles.) The value of the variable
Both Neptune and Pluto travel around the sun in elliptical orbits. For Neptune's orbit, a = 30.10, and for Pluto's orbit, a = 39.44, where the variable a represents their average distances from the sun in astronomical units. (One astronomical unit equals 93 million miles.) The value of the variable
Both Neptune and Pluto travel around the sun in elliptical orbits. For Neptune's orbit, a = 30.10, and for Pluto's orbit, a = 39.44, where the variable a represents their average distances from the sun in astronomical units. (One astronomical unit equals 93 million miles.) The value of the variable
Both Neptune and Pluto travel around the sun in elliptical orbits. For Neptune's orbit, a = 30.10, and for Pluto's orbit, a = 39.44, where the variable a represents their average distances from the sun in astronomical units. (One astronomical unit equals 93 million miles.) The value of the variable
Explain how the distance between the foci of an ellipse affects the shape of the ellipse.
Given the standard equation of an ellipse, explain how to determine the length of the major axis. How can you determine whether the major axis is vertical or horizontal?
Find the standard equation of a hyperbola with center (h, k) that satisfies the given conditions.Center (2,-2), focus (4, -2), and vertex (3,-2)
Find an equation of the parabola with vertex (0, 0) that satisfies the given conditions.Vertical axis, passing through (-2, 3)
Match the equation with its graph (a-d). I - h 6 z(1-4) (1+x)
Sketch a graph of the hyperbola, including the asymptotes. Give the coordinates of the vertices and foci. [= 9 16 z(1-x) (1 + 4)
An elliptical arch under a bridge is constructed so that it is 80 feet wide and has a maximum height of 30 feet, as illustrated in the figure. Find the height of the arch 10 feet from the center of the arch.
Find an equation of the parabola with vertex (0, 0) that satisfies the given conditions.Horizontal axis, passing through (1, -2)
Match the equation with its graph (a-d). (x + 1)² 4 4²- 1 9
Sketch a graph of the hyperbola, including the asymptotes. Give the coordinates of the vertices and foci. (x + 2)² (y + 1)² 4 16 1
The following graph shows one possible relationship between a person's weight and the effects of caffeine.According to this graph, does a single 12-ounce can of Mountain Dew contain enough caffeine for most people to avoid a headache? Caffeine
Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. x + 3y - 2z = 3 x+ 2 =-2 2 = 1 -x - 2y + 2x - 7y + +
Let A be the given matrix. Use technology to find det A. State whether A is invertible. 6 -7 23 -7 -1 3 3-4 54 77
Complete the following. (a) Write the system in the form AX= B. (b) Solve the system by finding A-1 and then using the equation X = A-1B. x + 2y = 3 x + 3y = 6
Use the given A and B to evaluate each expression.BA 3-2 4 2-3 5 4 A = 5 7 B = 11 -1 0-7 -6 4 3
Rooms at a hotel are regularly $110, but the cost of every room is reduced by $2 for each additional room rented. (a) Write a quadratic function C that gives the total cost of renting x rooms. (b) Solve C(x) = 1470 and interpret the result. (c) Find the absolute maximum for C and
A company is selling a product at market price that has a daily cost function C(x) = 7000x + 50,000 in dollars and a daily revenue function R(x) = 8000x in dollars, where x is units sold. (a) Determine the coordinates of the break-even point and interpret its meaning. (b) Graph the cost
Use the given A and B to evaluate each expression. 3-2 4 2-3 5 4 A = 5 7 B = 11 -1 0-7 -6 4 3
Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. 2x + 5y + 2 = 8 x + 2y - z = 2 3x + 7y = 5
Complete the following. (a) Write the system in the form AX= B. (b) Solve the system by finding A-1 and then using the equation X = A-1B. 2x + y = 4 -x + 2y = -1
Complete the following. (a) Write the system in the form AX= B. (b) Solve the system by finding A-1 and then using the equation X = A-1B. -x + 2y = 5 3x - 5y = -2
Use the given A and B to evaluate each expression. 3-2 4 2-3 5 4 A = 5 7 B = 11 -1 0-7 -6 4 3
A business has three machines that manufacture containers. Together they can make 100 containers per day, whereas the two fastest machines can make 80 containers per day. The fastest machine makes 34 more containers per day than the slowest machine. (a) Let x, y, and be the numbers of
A company is selling a product at market price that has a daily cost function C(x) = 0.05x2 + 300 in dollars and a daily revenue function R(x) = 12x in dollars, where x is units sold. (a) Determine the coordinates of the break-even points to the nearest hundredth. (b) Graph the cost and
Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. x + y + z = 3 x + y + 2z = 4 2x + 2y + 3z = 7
Use a graphing calculator to evaluate the expression with the given matrices A, B, and C. Compare your answers for parts (a) and (b). Then interpret the results. 414-643 3-5 B = 3 -4 -5 7 [14 -3 0-2 C = 81 46
The radius r in inches of a spherical balloon after 1 seconds is given by r = √t. (a) Is the radius increasing or decreasing? (b) Write a formula for a function V that calculates the volume of the sphere after 1 seconds. (c) Evaluate V(4) and interpret the result.
The formulaconverts degrees Fahrenheit to degrees Celsius. (a) Find f-1(x). (b) What does f-1 compute? f(x) = (x - 32) 9
Complete the following. (a) Write the system in the form AX= B. (b) Solve the system by finding A-1 and then using the equation X = A-1B. = x + 3y = 2x + 5y = −3 −2 -2 น
A company is selling a product at market price that has a daily cost function C(x) = x2 + 500 in dollars and a daily revenue function R(x) = 55x in dollars, where x is units sold. (a) Determine the coordinates of the break-even points to the nearest hundredth. (b) Graph the cost and
Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. -x + 2y + 4z = 10 3x - 2y - 2z = -12 x + 2y + 62 = 8
Approximate the radius r and height h of a cylindrical container with a volume V of 30 cubic inches and a lateral (side) surface area S of 45 square inches.
Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. 4x - 2y + 4z = 8 3x - 7y + 62 = 4 -x - 5y + 2z = 1
Use a graphing calculator to evaluate the expression with the given matrices A, B, and C. Compare your answers for parts (a) and (b). Then interpret the results. 414-643 3-5 B = 3 -4 -5 7 [14 -3 0-2 C = 81 46
A student takes out two loans totaling $2000 to help pay for college expenses. One loan is at 7% interest, and the other is at 9%. Interest for both loans is compounded annually. (a) If the combined total interest for the first year is $156, find the amount of each loan symbolically.(b)
The following graph shows one possible relationship between a person's weight and the effects of caffeine.What does the graph indicate about the effects of caffeine on 140-pound person who has consumed 335 mg of caffeine? Caffeine (mg) 350 300 250 200 150 100 50 0 Jitters 10c-7w=1740 5c-w =
Complete the following. (a) Write the system in the form AX = B. (b) Solve the system by finding A-1 and then using the equation X = A-1B. x + -7 -13 2x + y + 3z = -x + y + z = -4
When using elimination and substitution, explain how to recognize a system of linear equations that has no solutions.
The following graph shows one possible relationship between a person's weight and the effects of caffeine.Suppose a 180-pound person wishes to avoid both a headache and the jitters. What range of caffeine consumption is suggested? Caffeine (mg) 350 300 250 200 150 100 50 0 Jitters 10c-7w=1740 5c-w
There are initially 200,000 bacteria per milliliter in a sample. The number of bacteria reaches 300,000 per milliliter after 3 hours. (a) Use the formula N(1) = N0ekt to model the concentration of bacteria after / hours. (b) Evaluate N(5) and interpret the result. (c) After how long
When using elimination and substitution, explain how to recognize a system of linear equations that has infinitely many solutions.
The following graph shows one possible relationship between a person's weight and the effects of caffeine.For what weights could a person drink a 16-ounce can of All City NRG without experiencing the jitters? Caffeine (mg) 350 300 250 200 150 100 50 0 Jitters 10c-7w=1740 5c-w = 320 Headache 120 140
A music store marks its DVDs as A or B to indicate one of two selling prices. Each row in the table represents a purchase. Determine the cost of each type of DVD by using a matrix inverse. A B 1 2 2 3 Total $37.47 $61.95
The screen of a rectangular television set is 3 inches wider than it is high. If the perimeter of the screen is 42 inches, find its dimensions by writing a system of linear equations and solving.
Use a graphing calculator to evaluate the expression with the given matrices A, B, and C. Compare your answers for parts (a) and (b). Then interpret the results. 414-643 3-5 B = 3 -4 -5 7 [14 -3 0-2 C = 81 46
Complete the following. (a) Write the system in the form AX= B. (b) Solve the system by finding A-1 and then using the equation X = A-1B. -2x + y x -x + y + -5 = = -5
Determine a, b, and c so that the formula f(x) = ax2 + bx + c models the data in the table exactly. r f(x) -1 -8 1 6 3 -4
Complete the following for the given system of linear equations. (a) Write the system in the form AX = B. (b) Solve the linear system by computing X = A-1B with a calculator. Approximate the solution to the nearest hundredth when appropriate. 0.08x 0.7y= -0.504 1.1.x -0.05y = 0.73
Complete the following for the given system of linear equations. (a) Write the system in the form AX = B. (b) Solve the linear system by computing X = A-1B with a calculator. Approximate the solution to the nearest hundredth when appropriate. 64.1 31x + 18y 5x23y = -59.6
Find the maximum value of P = 3x + 4y subject to the following constraints. x + 3y $ 12 3x + y s 12 x 2 0,y ≥ 0
The following graph shows a weight and height chart. The weight w is listed in pounds and the height h in inches. The shaded area is a recommended region.Use the graph to find a system of linear inequalities region. that describes the recommended Height (inches) h KERRR
Complete the following for the given system of linear equations. (a) Write the system in the form AX = B. (b) Solve the linear system by computing X = A-1B with a calculator. Approximate the solution to the nearest hundredth when appropriate. 1.5x + 3.7y= 0.32 -0.4x 2.1y = 0.36 -

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