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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?
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) +
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
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
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) =
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
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
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
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).
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
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
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
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
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.
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
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
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
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
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.
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
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.
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
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.
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
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
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
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
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
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
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
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°.
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
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°.
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
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) =
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
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,
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
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
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
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,
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,
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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).
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

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