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mathematics
precalculus 1st

Precalculus 1st Edition Jay Abramson - Solutions

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists. 5x − y = 1−10x + 2y = − 2
Can a matrix with an entire column of zeros have an inverse? Explain why or why not.
Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you had we eventually simplify to 7x + 13 = A(3x + 5) + B(x+1). Explain how you could intelligently choose an x-value that will eliminate either A or B and
The determinant of 2×2 matrix A is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.
For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists. 4x - 6y - 2z = 1 10 x - 7y + 5z = - 3x + 6 - 92 = 65 1 4
Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.
Does matrix multiplication commute? That is, does AB = BA? If so, prove why it does. If not, explain why it does not.
Can a matrix that has 0 entries for an entire row have one solution? Explain why or why not.
For the following exercises, use a graphing utility to graph the given parametric equations.a.b.c.Graph all three sets of parametric equations on the domain [−4π, 6π]. [x(t) = cost-1 y(t) = sin t + t
For the following exercises, use a graphing utility to graph the given parametric equations.a.b.c.Graph all three sets of parametric equations on the domain [0, 2π]. [x(t) = cost-1 y(t) = sin t + t
If you perform your break-even analysis and there is more than one solution, explain how you would determine which x-values are profit and which are not.
Given a system of equations, explain at least two different methods of solving that system.
For the following exercises, use a graphing utility to graph the given parametric equations.a.b.c.Graph all three sets of parametric equations on the domain [0, 4π]. [x(t) = cost-1 y(t) = sin t + t
For the following exercises, use a graphing utility to graph the given parametric equations.a.b.c.Explain the effect on the graph of the parametric equation when we switched sin t and cos t. [x(t) = cost-1 y(t) = sin t + t
For the following exercises, use a graphing utility to graph the given parametric equations.a.b.c.The graph of each set of parametric equations appears to “creep” along one of the axes. What controls which axis the graph creeps along? [x(t) = cost-1 y(t) = sin t + t
For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain. [x(t) = 6sin t+2sin(6t) An
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.r1 = √3 , r2 = 2sin(θ)
For the following exercises, find the area of the triangle.Los Angeles is 1,744 miles from Chicago, Chicago is 714 miles from New York, and New York is 2,451 miles from Los Angeles. Draw a triangle connecting these three cities, and find the angles in the triangle.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Use a graphing calculator to find the rectangular coordinates of (−3, 3π/7). Round to the nearest thousandth.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.As part of a video game, the point (5,7) is rotated counterclockwise about the origin through an angle of 35°. Find the new coordinates of this point.
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.A street light is mounted on a pole. A 6-foot-tall man is standing on the street a short distance from the pole, casting a shadow. The angle of elevation from the tip of the man’s shadow to the top
For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain. 2sin An hypotrochoid: {(t)=5cost
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.r1 2 = sin θ, r2 2 = cos θ
For the following exercises, find the area of the triangle.Philadelphia is 140 miles from Washington, D.C., Washington, D.C. is 442 miles from Boston, and Boston is 315 miles from Philadelphia. Draw a triangle connecting these three cities and find the angles in the triangle.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Use a graphing calculator to find the polar coordinates of (−7, 8) in degrees. Round to the nearest thousandth.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.As part of a video game, the point (7,3) is rotated counterclockwise about the origin through an angle of 40°. Find the new coordinates of this point.
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.Three cities, A, B, and C, are located so that city A is due east of city B. If city C is located 35° west of north from city B and is 100 miles from city A and 70 miles from city B, how far is city
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.r1 = 1 + cos θ, r2 = 1 − sin θ
For the following exercises, find the area of the triangle.Two planes leave the same airport at the same time. One flies at 20° east of north at 500 miles per hour. The second flies at 30° east of south at 600 miles per hour. How far apart are the planes after 2 hours?
For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain. x(t) = 5sin(2t) sin t y(t) = 5
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Use a graphing calculator to find the polar coordinates of (3,−4) in degrees. Round to the nearest hundredth.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.Two streets meet at an 80° angle. At the corner, a park is being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road,
For the following exercises, find the area of the triangle.Two airplanes take off in different directions. One travels 300 mph due west and the other travels 25° north of west at 420 mph. After 90 minutes, how far apart are they, assuming they are flying at the same altitude?
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Use a graphing calculator to find the polar coordinates of (−2, 0) in radians. Round to the nearest hundredth.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Describe the graph of r = asec θ; a > 0.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.
For the following exercises, find the area of the triangle.A parallelogram has sides of length 15.4 units and 9.8 units. Its area is 72.9 square units. Find the measure of the longer diagonal.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Describe the graph of r = asec θ; a < 0.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.A 50-pound object rests on a ramp that is inclined 19°. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.The Bermuda triangle is a region of the Atlantic Ocean that connects Bermuda, Florida, and Puerto Rico. Find the area of the Bermuda triangle if the distance from Florida to Bermuda is 1030 miles,
For the following exercises, find the area of the triangle.The four sequential sides of a quadrilateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 12.9 cm. The angle between the two smallest sides is 117°. What is the area of this quadrilateral?
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.Naomi bought a modern dining table whose top is in the shape of a triangle. Find the area of the table top if two of the sides measure 4 feet and 4.5 feet, and the smaller angles measure 32° and
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Describe the graph of r = acsc θ; a > 0.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it upward, and 5 pounds acting on it 45° from the horizontal. What single force is the
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.A yield sign measures 30 inches on all three sides. What is the area of the sign?
For the following exercises, find the area of the triangle.The four sequential sides of a quadrilateral have lengths 5.7 cm, 7.2 cm, 9.4 cm, and 12.8 cm. The angle between the two smallest sides is 106°. What is the area of this quadrilateral?
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Describe the graph of r = acsc θ; a < 0.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it −135° from the horizontal, and 5 pounds acting on it directed 150° from the
For the following exercises, find the area of the triangle.Find the area of a triangular piece of land that measures 30 feet on one side and 42 feet on another; the included angle measures 132°. Round to the nearest whole square foot.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.What polar equations will give an oblique line?
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 2 pounds acting on it to the right, 5 pounds
For the following exercises, find the area of the triangle.Find the area of a triangular piece of land that measures 110 feet on one side and 250 feet on another; the included angle measures 85°. Round to the nearest whole square foot.
For the following exercises, graph the polar inequality.r < 4
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed
For the following exercises, graph the polar inequality.0 ≤ θ ≤ π/4
For the following exercises, graph the polar inequality.θ = π/4, r ≥ 2
For the following exercises, use a graphing utility to graph the given parametric equations.a.b.c.Explain the effect on the graph of the parametric equation when we changed the domain. [x(t) = cost-1 y(t) = sin t + t
For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of [0,4π] and then explain the differences shown in the graphs.r = sin(cos(3θ)) r = sin(3θ)
For the following exercises, find the area of the triangle.A satellite calculates the distances and angle shown in Figure 15 (not to scale). Find the distance between the two cities. Round answers to the nearest tenth. 370 km Figure 15 2.1° 350 km
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.r = −4
For the following exercises, calculate u ⋅ v.Given initial point P1 = (3, 2) and terminal point P2 = (−5,−1), write the vector v in terms of i and j. Draw the points and the vector on the graph.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.Similar to an angle of elevation, an angle of depression is the acute angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal. A pilot is flying over a
For the following exercises, use a graphing utility to graph the given parametric equations.A skateboarder riding on a level surface at a constant speed of 9 ft/s throws a ball in the air, the height of which can be described by the equation y(t) = −16t2  + 10t + 5.Write parametric
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection. r1 = 3 + 2sin θ, r2 = 2
For the following exercises, find the area of the triangle.An airplane flies 220 miles with a heading of 40°, and then flies 180 miles with a heading of 170°. How far is the plane from its starting point, and at what heading? Round answers to the nearest tenth.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.θ = −2π/3
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 4.3 km apart, to be 32° and 56°, as shown in Figure 33. Find the distance of the plane from point
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.A man starts walking from home and walks 3 miles at 20° north of west, then 5 miles at 10° west of south, then 4 miles at 15° north of east. If he walked straight home, how
For the following exercises, find the area of the triangle.A 113-foot tower is located on a hill that is inclined 34° to the horizontal, as shown in Figure 16. A guywire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of
For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time t. Neglect air resistance. Find parametric equations that model the problem situation.
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.r1 = 6 − 4cos θ, r2 = 4
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.θ = π/4
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.A woman starts walking from home and walks 6 miles at 40° north of east, then 2 miles at 15° east of south, then 5 miles at 30° south of west. If she walked straight home,
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top
For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°.Consider the position of the dart at any time t. Neglect air resistance.Find all possible values of x that represent the situation.
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.r1 = 1 + sin θ, r2 = 3sin θ
For the following exercises, find the area of the triangle.Two ships left a port at the same time. One ship traveled at a speed of 18 miles per hour at a heading of 320°. The other ship traveled at a speed of 22 miles per hour at a heading of 194°. Find the distance between the two ships after 10
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.r = sec θ
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing from the southwest at 80 km/hr. How many degrees off course will the plane end up flying,
For the following exercises, find the area of the triangle.The graph in Figure 17 represents two boats departing at the same time from the same dock. The first boat is traveling at 18 miles per hour at a heading of 327° and the second boat is traveling at 4 miles per hour at a heading of 60°.
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top
For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°.Consider the position of the dart at any time t. Neglect air resistance.When will the dart hit the ground?
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.r1 = 1 + cos θ, r2 = 3cos θ
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.r = −10sin θ
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.An airplane is heading north at an airspeed of 500 km/hr, but there is a wind blowing from the northwest at 50 km/hr. How many degrees off course will the plane end up flying,
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.Points A and B are on opposite sides of a lake. Point C is 97 meters from A. The measure of angle BAC is determined to be 101°, and the measure of angle ACB is determined to be 53°. What is the
For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°.Consider the position of the dart at any time t. Neglect air resistance.Find the maximum height of the dart.
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.r1 = cos(2θ), r2 = sin(2θ)
For the following exercises, find the area of the triangle.A triangular swimming pool measures 40 feet on one side and 65 feet on another side. These sides form an angle that measures 50°. How long is the third side (to the nearest tenth)?
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.r = 3cos θ
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km/hr.The plane flies with an airspeed of 550 km/hr. To end up flying due north, how
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.A man and a woman standing 3 1/2 miles apart spot a hot air balloon at the same time. If the angle of elevation from the man to the balloon is 27°, and the angle of elevation from the woman to the
For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°.Consider the position of the dart at any time t. Neglect air resistance.At what time will the dart reach maximum height?
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.r1 = sin2 (2θ), r2 = 1 − cos(4θ)
For the following exercises, find the area of the triangle.A pilot flies in a straight path for 1 hour 30 min. She then makes a course correction, heading 10° to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 680 miles per hour, how
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.Use a graphing calculator to find the rectangular coordinates of (2, −π/5). Round to the nearest thousandth.
For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.An airplane needs to head due north, but there is a wind blowing from the northwest at 80 km/hr. The plane flies with an airspeed of 500 km/hr. To end up flying due north, how
For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain. [x(t) = 14cost-cos(141) An

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