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Discovering Advanced Algebra An Investigative Approach 1st edition Jerald Murdock, Ellen Kamischke, Eric Kamischke - Solutions

A ship's captain sees a lighthouse at a bearing of 105°. Then the captain sails 8.8 nautical miles on a bearing of 174°. The lighthouse is now at a bearing of 52°. How far is the ship from the lighthouse?
Here are the batting averages of the National League's Most Valuable Players from 1980 to 2000. (The New York Times Almanac 2002) {.286, .316, .281, .302, .314, .353, .290, .287, .290, .291, .301, .319, .311, .336, .368, .319, .326, .366, .308, .319, .334} a. Make a box plot of these data. b. Give
Find the total surface area of the figure at right. Round your answer to the nearest square centimeter.
Congruent circles A and B are tangent at point C. The radius of each circle is 3 units. Rays RU and ST are tangent at U and S, respectively, and intersect at T. Find ST.
Find the measure of (T.
Solve for A and b. (Assume b is positive.) a. 16 = 25 + 36 - 2(5)(6) cos A b. 49 = b2 + 9 - 2(3)(b) cos 60°
Two airplanes pass over Chicago, Illinois, at the same time. One is cruising at 400 mi/h on a bearing of 105°, and the other is cruising at 450 mi/h on a bearing of 260°. How far apart will they be after 2 h?
Find the measure of (S.
Seismic exploration identifies underground phenomena, such as caves, oil pockets, and rock layers, by transmitting sound into the earth and timing the echo of the vibration. From a sounding at point A, a "thumper" truck locates an underground chamber 7 km away. Moving to point B, 5 km from point A,
A folding chair's legs meet to form a 50° angle. The rear leg is 55 cm long and attaches to the front leg at a point 75 cm from the front leg's foot. How far apart are the legs at the floor?
The Hear Me Now Phone Company plans to build a cell tower to serve the needs of Pleasant Beach and the beachfront. It decides to locate the cell tower so that Pleasant Beach is 1 mi away at a bearing of 60° from the tower. The range of the signal from the cell tower is 1.75 mi. The beachfront
Use the parametric equations x = - 3t + 1 and y = 2/t + 1 to answer each question. a. Find the x- and y-coordinates of the points that correspond to the values of t = 3, t = 0, and t = - 3. b. Find the y-value that corresponds to an x-value of - 7. c. Find the x-value that corresponds to a y-value
A pilot is flying to a destination 700 mi away at a bearing of 105°. The cruising speed of the plane is 500 mi/h, and the wind is blowing between 20 mi/h and 30 mi/h at a bearing of 30°. At what bearing should she aim the plane to compensate for the wind?
Use the Law of Sines and the Law of Cosines to find all of the missing side lengths and angle measures.a.b.
A raft is being moved by a wind blowing east at 20 m/s and a current flowing south at 30 m/s. a. What is the raft's position after 8 s relative to its starting position? b. What equations simulate this motion?
Graph each pair of parametric equations in a friendly window with factor 2, then eliminate the parameter to get a single equation using only x and y. Graph the resulting equation and describe how it compares with the original graph. a. x = 2t - 5 and y = t + 1 b. x = t2 + 1 and y = t - 2, - 2 ( t (
Write parametric equations that will result in each transformation below for the equations x = 2t - 5 and y = t + 1. a. Reflect the curve across the y-axis. b. Reflect the curve across the x-axis. c. Translate the curve up 3 units. d. Translate the curve left 4 units and down 2 units.
For each triangle, find the measure of the labeled angle or the length of the labeled side.a.b. c. d. e. f.
Sketch a graph of x = t cos 28° and y = t sin 28°. What is the angle between the graph and the x-axis?
A diver runs off a 10 m platform with an initial horizontal velocity of 4 m/s. The edge of the platform is directly above a point 1.5 m from the pool's edge. Simulate her motion on your graphing calculator. How far from the edge of the pool will she hit the water?
A duck paddles at a rate of 2.4 ft/s, aiming directly for the opposite bank of a 47 ft wide river. When he lands, he finds himself 28 ft downstream from the point where he started. What is the speed of the current?
Aliya sees a coconut 66 ft up in a coconut palm that is 20 ft away from her. She has a slingshot capable of launching a rock at 100 ft/s. If she launches a rock at an angle of 72°, will she hit the coconut? If not, by how much will she miss it?
1. In previous chapters you made connections between equations and their graphs. For example, the graph of an equation in the form y = a + bx is a line, and the graph of an equation in the form y = (x2 + bx + c is a parabola. Can you make similar connections for parametric equations? What type of
Find the distance between each pair of points. a. (2, 5) and (8, 13) b. (0, 3) and (5, 10) c. (- 4, 6) and (- 2, - 3) d. (3, d) and (- 6, 3d)
A 24 ft ladder is placed upright against a wall. Then the top of the ladder slides down the wall while the foot of the ladder is pulled outward along the ground at a steady rate of 2 ft/s.a. Find the heights that the ladder reaches at 1 s intervals while the ladder slides down the wall.b. How long
Let d represent the distance between the point (5, - 3) and any point, (x, y), on the parabola y = 0.5x2 + 1.a. Write an equation for d in terms of x. b. What is the minimum distance? What are the coordinates of the point on the parabola that is closest to the point (5, - 3)?
The city councils of three neighboring towns-Ashton, Bradburg, and Carlville-decide to pool their resources and build a recreation center. To be fair, they decide to locate the recreation center equidistant from all three towns. a. When a coordinate plane is placed on a map of the towns, Ashton is
Follow these steps to solve Exercise 12a with geometry software. a. Open a new sketch. Define a coordinate system and plot three points, A, B, and C, that represent the locations of Ashton, Bradburg, and Carlville. b. Connect the three points with line segments. Construct the perpendicular bisector
Complete the square in each equation such that the left side represents a perfect square or a sum of perfect squares. a. x2 + 6x = 5 b. y2 - 4y = - 1 c. x2 + 6x + y2 - 4y = 4
Give the domain and range of the function f (x) = x2 + 6x + 7.
A ship leaves port and travels on a bearing of 205° for 2.5 h at 8 knots, and then on a bearing of 150° for 3 h at 10 knots. How far is the ship from its port? (A knot is equivalent to 1 nautical mile per hour.)
A sealed 10 cm tall cone resting on its base is filled to half its height with liquid. It is then turned upside down. To what height, to the nearest hundredth of a centimeter, does the liquid reach?
The distance between the points (2, 7) and (5, y) is 5 units. Find the possible value(s) of y.
The distance between the points (- 1, 5) and (x, - 2) is 47. Find the possible value(s) of x.
Which side is longest in the triangle with vertices A(1, 2), B(3, - 1), and C(5, 3)?
Find the perimeter of the triangle with vertices A(8, - 2), B(1, 5), and C(4, - 5).
Find the equation of the locus of points that are twice as far from the point (2, 0) as they are from (5, 0).
If you are too close to a radio tower, you will be unable to pick up its signal. Let the center of a town be represented by the origin of a coordinate plane. Suppose a radio tower is located 2 mi east and 3 mi north of the center of town, or at the point (2, 3). A highway runs north-south 2.5 mi
Josh is riding his mountain bike when he realizes that he needs to get home quickly for dinner. He is 2 mi from the road, and home is 3 mi down that road. He can ride 9 mi/h through the field separating him from the road and can ride 22 mi/h on the road.a. If Josh rides through the field to a point
A 10 m pole and a 13 m pole are 20 m apart at their bases. A wire connects the top of each pole with a point on the ground between them.a. Let y represent the total length of the wire. Write an equation that relates x and y.b. What domain and range make sense in this situation?c. Where should the
Sketch each circle on your paper, and label the center and the radius. For a-d, rewrite the equation as two functions. Use your calculator to check your work. a. x2 + y2 = 4 b. (x - 3)2 + y2 = 1 c. (x + 1)2 + (y - 2)2 = 9 d. x2 + (y - 1.5)2 = 0.25 e. x = 2 cos t + 1 y = 2 cos t + 2 f. x = 4 cos t -
Read the connection below about the reflection property of an ellipse. If a room is constructed in the shape of an ellipse, and you stand at one focus and speak softly, a person standing at the other focus will hear you clearly. Such rooms are often called whispering chambers. Consider a whispering
One possible gear ratio on Matthew's mountain bike is 4 to 1. This means that the front gear has four times as many teeth as the gear on the back wheel. So each revolution of the pedal causes the rear wheel to make four revolutions.a. If Matthew is pedaling 60 revolutions per minute (r/min), how
The Moon's greatest distance from Earth is 252,710 mi, and its smallest distance is 221,643 mi. Write an equation that describes the Moon's orbit around Earth. Earth is at one focus of the Moon's elliptical orbit.
Solve a system of equations and find the quadratic equation, y = ax2 + bx + c, that fits these data points.
Find the perimeter of this triangle.
Write an equation in standard form for each graph.a.b. c. d.
Write parametric equations for each graph in Exercise 2.a.b. c. d.
Write parametric equations for each circle described. a. Radius 2, center (0, 3) b. Radius 6, center (- 1, 2)
Write an equation in standard form for each graph.a.b. c. d.
Find the exact coordinates of the foci for each ellipse in Exercise 6.a.b. c. d.
Suppose you placed a grid on the plane of a comet's orbit, with the origin at the sun and the x-axis running through the longer axis of the orbit, as shown in the diagram. The table gives the approximate coordinates of the comet as it orbits the Sun. Both x and y are measured in astronomical units
The top of a doorway is designed to be half an ellipse. The width of the doorway is 1.6 m, and the height of the half-ellipse is designed to be 62.4 cm. The crew have nails and string available. They want to trace the half-ellipse with a pencil before they cut the plywood to go over the doorway. a.
For each parabola described, use the information given to find the location of the missing feature. It may help to draw a sketch. a. If the focus is (1, 4), and the directrix is y = - 3, where is the vertex? b. If the vertex is (- 2, 2), and the focus is (- 2, - 4), what is the equation of the
Sheila is designing a parabolic dish to use for cooking on a camping trip. She plans to make the dish 40 cm wide and 20 cm deep. Where should she locate the cooking grill so that all of the light that enters the parabolic dish will be reflected toward the food?
The diagram at right shows the reflection of a ray of light in a parabolic reflector. The angles A and B are equal. Follow these steps to verify this property of parabolas.a. Sketch the parabola y2 = 8x.b. What are the coordinates of the focus of this parabola? c. On the same graph, sketch the line
Find the equation that describes a parabola containing the points (3.6, 0.764), (5, 1.436), and (5.8, - 2.404).
Find the minimum distance from the origin to the parabola y = - x2 + 1. What point(s) on the parabola is closest to the origin?
Find the equation of the ellipse with foci (- 6, 1) and (10, 1) that passes through the point (10, 13).
Consider the polynomial function f(x) = 2x3 - 5x2 + 22x - 10. a. What are the possible rational roots of f (x)? b. Find all rational roots. c. Write the equation in factored form.
On a three-dimensional coordinate system with variables x, y, and z, the standard equation of a plane is in the form ax + by + cz = d. Find the intersection of the three planes described by 3x + y + 2z = - 11, - 4x + 3y + 3z = - 2, and x - 2y - z = - 3.
Sketch each parabola, and label the vertex and line of symmetry. a. (π/2)2 + 5 = y b. (y + 2)2 - 2 = x c. -(x + 3)2 + 1 = 2y d. 2y2 = - x + 4 e. x = 4t - 1 y = 2t2 + 3 f. x = 3t2 + 2 y = 5t
Write an equation in standard form for each parabola.a.b. c. d.
Write parametric equations for each parabola in Exercise 4.a.b. c. d.
Find the equation of the parabola with directrix x = 3 and vertex (0, 0).
The pilot of a small boat charts a course such that the boat will always be equidistant from an upcoming rock and the shoreline. Describe the path of the boat. If the rock is 2 miles offshore, write an equation for the path of the boat.
Consider the graph at right.a. Because d1 = d2, you can write the equationRewrite this equation by solving for y. b. Describe the graph represented by your equation from 8a.
Write the equation of the parabola with focus (1, 3) and directrix y = - 1.
Sketch each hyperbola on your paper. Write the coordinates of each vertex and the equation of each asymptote.a.b. c. d. e. x = 4/cos t -1 y = 2 tan t + 3 f. x = 3 tan t + 3 y = 5/cos t
A receiver can determine the distance to a homing transmitter by its signal strength. These signal strengths were measured using a receiver in a car traveling due north.a. Find the equation of the hyperbola that best fits the data. b. Name the center of this hyperbola. What does this point tell
Sketch the graphs of the conic sections in 11a-d. a. y = x2 b. x2 + y2 = 9 c. x2/9 + y2/16 = 1 d. x2/9 - y2/16 = 1 e. If each of the curves in 11a-d is rotated about the y-axis, describe the shape that is formed. Include a sketch.
Find the vertical distance between a point on the hyperbola (y + 1/2)2 - (x - 2/3)2 = 1 and its nearest asymptote for each x-value shown at right.
Solve the quadratic equation 0 = - x2 + 6x - 5 by completing the square.
Mercury's orbit is an ellipse with the Sun at one focus, eccentricity 0.206, and major axis approximately 1.158 × 108 km. If you consider Mercury's orbit with the Sun at the origin and the other focus on the positive x-axis, what equation models the orbit?
The setter on a volleyball team makes contact with the ball at a height of 5 ft. The parabolic path of the ball reaches a maximum height of 17.5 ft when the ball is 10 ft from the setter. a. Find an equation that models the ball's path. b. A hitter can spike the ball when it is 8.5 feet off the
Sketch the graph of each parabola. Give the coordinates of each vertex and focus, and the equation of each directrix. a. y = - (x + 1)2 - 2 b. y = 1/2 x2 - 3x + 5 c. x = 1/2 t2 - 6 y = t
The half-life of radium-226 is 1620 yr. a. Write a function that relates the amount s of a sample of radium-226 remaining after t years. b. After 1000 yr, how much of a 500 g sample of radium-226 will remain? c. How long will it take for a 3 kg sample of radium-226 to decay so that only 10 g
What are the coordinates of the foci of each hyperbola in Exercise 1?a.b. c. d. e. x = 4/cos t -1 y = 2 tan t + 3 f. x = 3 tan t + 3 y = 5/cos t
Write an equation in standard form for each graph.a.b. c. d.
Write parametric equations for each graph in Exercise 3.a.b. c. d.
Write the equations of the asymptotes for each hyperbola in Exercise 3.a.b. c. d.
Another way to locate the foci of a hyperbola is by rotating the asymptote rectangle about its center so that opposite corners lie on the line of symmetry that contains the vertices of the hyperbola. From the diagram, you can see that the distance from the origin to a focus is one-half the length
A point moves in a plane so that the difference of its distances from (- 5, 1) and (7, 1) is always 10 units. What is the equation of the path of this point?
Graph and write the equation of a hyperbola that has an upper vertex at (- 2.35, 1.46) and has an asymptote of y = 1.5x + 1.035.
Approximate the equation of each hyperbola shown.a.b.
Rewrite each equation in the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. a. (x + 7)2 = 9(y - 11) b. (x - 7)2/9 + (y + 11)2/1 = 1 c. (x - 1)2 + (y + 3)2 = 5 d. (x - 2)2/4 - (y + 3)2/9 = 1
Three LORAN radio transmitters, A, B, and C, are located 200 miles apart along a straight coastline. They simultaneously transmit radio signals at regular intervals. The signals travel at a speed of 980 feet per microsecond. A ship, at S, first receives a signal from transmitter B. After 264
Find the equation of the circle that passes through the four intersection points of the ellipses x2/16 + y2/9 = 1 and x2/9 + y2/16 = 1.
Find the equations of two parabolas that pass through the points (2, 5), (0, 9), and (- 6, 7). Sketch each parabola.
Find the coordinates of the foci of each ellipse. a. (x + 2/4)2 + (y - 5/6)2 = 1 b. x = cos t + 1 y = 0.5 sin t - 2
Find the equations of the asymptotes of the hyperbola with vertices (5, 8.5) and (5, 3.5), and foci (5, 12.5) and (5, - 0.5).
If the vertices of a triangle are A(10, 16), B(4, 9), and C(8, 1), find m∠ABC.
Write the polynomial expression x4 - 3x3 + 4x2 - 6x + 4 in factored form.
You have seen that a double cone can be intersected with a plane to form a circle, an ellipse, a parabola, and a hyperbola. What shapes can be formed by the intersection of a plane and a square-based pyramid? Draw a sketch of each possibility.
Point C is rotated 60° counterclockwise about the origin. Find the coordinates of the new point.
The parametric equations of a parabola are x = 3t2 + 3 and y = 5t. Rotate the parabola 135° counterclockwise. Write the parametric equations for the new parabola. Verify your results by graphing the original parabola and its rotated image.
Find the values for a, b, c, and d as you follow these steps to complete the square for 15x2 + 21x. 15x2 + 21x 15(x2 + ax) 15(x2 + ax + b) - 15b 15(x2 + ax + b) - c 15(x + d)2 - c
Rotate the hyperbola x2 - y2 = 1 counterclockwise 45° about the origin. Verify your result by graphing the original hyperbola and its rotated image.
Consider the parametric equations x1 = tan t and y1 = 1/ cos t. a. Predict what the graph will look like. b. Predict what the graph of x2 = x1 cos θ - y1 sin θ and y2 = x1 sin + y1 cos θ will be when equals 50°. c. Draw the graphs to check your predictions.
Convert each equation to the standard form of a conic section. Name the shape described by each equation. a. x2 - y2 + 8x + 10y + 2 = 0 b. 2x2 + y2 - 12x - 16y + 10 = 0 c. 3x2 + 30x + 5y - 4 = 0 d. 5x2 + 5y2 + 20x - 6 = 0

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