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linear algebra

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

Sketch a graph of each function. Label all zeros and the coordinates of all maximum and minimum points. (Each coordinate should be accurate to the nearest hundredth.) a. y = 2(x - 2)2 - 16 b. y = -3(x - 5)(x + 1) c. y = x2 - 3x + 2 d. y = (x + 1)(x - 3)(x + 4) e. y = x3 + 2x2 - 19x + 20 f. y = 2x5
Write the equation of each graph in factored form.a.b. c. d.
By postal regulations, the maximum combined girth and length of a rectangular package sent by Priority Mail is 108 in. The length is the longest dimension, and the girth is the perimeter of the cross section. Find the dimensions of the package with maximum volume that can be sent through the mail.
An object is dropped from the top of a building into a pool of water at ground level. There is a splash 6.8 s after the object is dropped. How high is the building in meters? In feet?
Consider this puzzle:a. Write a formula relating the greatest number of pieces of a circle, y, you can obtain from x cuts.b. Use the formula to find the maximum number of pieces with five cuts and with ten cuts.
Create a table for each equation with t = {-2, -1, 0, 1, 2}. a. x = 3t - 1 y = 2t + 1 b. x = t + 1 y = t2 c. x = t2 y = t + 3 d. x = t - 1 y = (4 - t2
Los Angeles, California, and Honolulu, Hawaii, are about 2500 mi apart. One plane flies from Los Angeles to Honolulu, and a second plane flies in the opposite direction.a. Describe the meaning of each number in the x-equations. (The equations for y simply assign noncolliding flight paths.)b. Use
These parametric equations simulate two walkers, with x and y measured in meters and t measured in seconds.a. Graph their motion for 0 ( t ( 5. b. Give real-world meanings for the values of 1.4, 3.1, 4.7, and 1.2 in the equations. c. Where do the two paths intersect? d. Do the walkers collide? How
An elephant stomps the ground creating sound waves in the air and vibrations (waves) in the ground. Sound waves travel through the air at approximately 0.3 km/s, and a vibration of the Earth's crust travels at approximately 6.1 km/s. a. Write parametric equations that model both waves for 3 s, and
Solve this system.
Find the exact roots of this equation without using a calculator. 0 = x3 - 4x2 + 2x + 4
Consider this sequence. -6, -4, 3, 15, 32( ( ( ( a. Find the nth term using finite differences. Assume -6 is u1. b. Find u20.
Find the equation of the line through (-3, 10) and (6, -5).
Find the equation of the parabola that passes through the points (-2, -20), (2, 0), and (4, -14).
Graph each pair of parametric equations on your calculator. Sketch the result and use arrows to indicate the direction of increasing t-values along the graph. Limit your t-values as indicated, or, if an interval for t isn't listed, then find one that shows all of the graph that fits in a friendly
Explore the graphs described in 3a-e, using a friendly window with a factor of 2 and -10 ( t ( 10. a. Graph x = t and y = t2. b. Graph x = t + 2 and y = t2. How does this graph compare with the graph in 3a? c. Graph x = t and y = t2 - 3. How does this graph compare with the graph in 3a? d. Predict
Explore these graphs, using a friendly window with a factor of 2 and -10 ( t ( 10. a. Graph x = t and y = |t|. b. Graph x = t - 1 and y = |t| + 2. How does this graph compare with the graph in 4a? c. Write a pair of parametric equations that will translate the graph in 4a left 4 units and down 3
The graphs below display a simulation of a team's mascot walking toward the west-end goal on the high school football field. Starting at a point 65 yards (yd) from the goal line and 50 ft from the sideline, she moves toward the goal line and waves at the crowd when she is 35 yd from the goal line.
Write parametric equations for each graph. (You can create parametric equations from a single equation that uses only x and y by letting x = t and changing y to a function of t by replacing x with t.)a.b. c. d.
The graph of the parametric equations x = ((t) and y = g(t) is pictured at right.a. Sketch a graph of x = ((t) and y = -g(t) and describe the transformation.b. Sketch a graph of x = -( (t) and y = g(t) and describe the transformation.
The graph of the parametric equations x = r(t) and y = s(t) is pictured at right. Write the parametric equations for each graph below in terms of r(t) and s(t).a.b. c. d.
Each spring a hare challenges a tortoise to a 50 m race. The hare knows that he can run faster than the tortoise, and he boasts that he can still win the race even if he gives the tortoise a 100 s head start. The tortoise crawls at a rate of 0.4 m/s, and the hare's running speed is 1.8 m/s. a.
1. In the investigation you used the parametric equations for a circle, x = 3cos t and y = 3 sin t, to graph other geometric shapes on your calculator. Mathematically, do you think it is correct to say that x = 3 cos t and y = 3 sin t are the parametric equations of a square or a hexagon? Explain
a. x = t + 1 b. x = 3t - 1 c. x = t2 d. x = t - 1
Write parametric equations for two perpendicular lines that intersect at the point (3, 2), with one line having a slope of -0.5.
The functions d1 = 1.5t and d2 = 12 - 2.5t represent Edna's and Maria's distances in miles from a trailhead, as functions of time in hours. a. Write parametric equations to simulate Edna's hike north away from the trailhead. Use x = 1. b. Write parametric equations to simulate Maria's hike south
An egg is dropped from the roof of a 98 m building. a. How long will it take the egg to reach the ground? b. Write and graph parametric equations to model the motion. c. When will the egg be 1.75 m above the ground? d. A 1.75 m high trampoline is rolled at a rate of 1.2 m/s toward the egg drop
At right are the graphs ofWrite parametric equations of the parabola's reflection across the line y = x.
Tanker A moves at 18 mi/h and Tanker B moves at 22 mi/h. Both are traveling from Corpus Christi, Texas, to St. Petersburg, Florida, 900 mi directly east. Simulate the tanker movements if Tanker A leaves Corpus Christi at noon and Tanker B leaves at 5 P.M. a. Write parametric equations to simulate
Consider the function f (x) = 3 + ((x - 1)2. a. Find i. ( (9) ii. ( (1) iii. ( (0) iv. ( (-7) b. Find the equation(s) of the inverse of ( (x). Is the inverse a function? c. Describe how you can use your calculations in 16a to check your inverse in 16b. d. Use your calculator to graph ( (x) and its
Write each pair of parametric equations as a single equation using only x and y. Graph this new relation in a friendly graphing window. Verify that the graph of the new equation is the same as the graph of the pair of parametric equations. a. x = t + 1 y = t2 b. x = 3t - 1 y = 2t + 1 c. x = t2 y =
Write a single equation (using only x and y) that is equivalent to each pair of parametric equations. a. x = 2t - 3 y = t + 2 b. x = t2 y = t + 1 c. x = 1/2t + 1 y = t - 2 / 3 d. x = t - 3 y = 2(t - 1)2
The table at right gives x- and y-values for several values of t.a. Write an equation for x in terms of t. b. Write an equation for y in terms of t. c. Eliminate the parameter and combine the equations in 4a and 4b. Verify that this equation fits the values of (x, y).
Use the graphs of x = ( (t) and y = g(t) to create a graph of y as a function of x.
Write parametric equations for x = ( (t) and y = g(t) given in Exercise 5, and an equation for y as a function of x. How do the slopes of these equations compare?
Consider the parametric equations x = ( (t) = t + 2 and y = g(t) = (1 - t2. a. Graph x = ( (t) and y = g(t). b. Graph x = ( (t) and y = -g(t) and identify the transformations of the original equations. Eliminate the parameter to write a single equation using only x and y. c. Graph x = -( (t) and y
A bug is crawling up a wall with locations given in the table at right. The variables x and y represent horizontal and vertical distances from the lower left corner of the wall, measured in inches, and t represents time measured in seconds.a. Write parametric equations for x and y, in terms of t,
Write all the trigonometric formulas (including inverses) relating the sides and angles in this triangle. There should be a total of 12.
Tanker A is moving at a speed of 18 mi/h from Corpus Christi, Texas, toward Panama City, Florida. Panama City is 750 mi from Corpus Christi at a bearing of 73°. a. Make a sketch of the tanker's motion, including coordinate axes. b. How long does the tanker take to get to Panama City? c. How far
Tanker B is traveling at a speed of 22 mi/h from St. Petersburg, Florida, to New Orleans, Louisiana, on a bearing of 285°. The distance between the two ports is 510 mi. a. Make a sketch of the tanker's motion, including coordinate axes. b. How long will it take the tanker to get to New Orleans? c.
Civil engineers generally bank, or angle, a curve on a road so that a car going around the curve at the recommended speed does not skid off the road. Engineers use this formula to calculate the proper banking angle, , where v represents the velocity in meters per second, r represents the radius of
This table gives the position of a walker at several times.a. Write a single equation for y in terms of x that fits the data points, (x, y), listed in the table. b. Parametric equations modeling this table are x = 6 + 4t and y = 5 + 3t. Eliminate the parameter, t, from these equations and compare
Graph the parabola y = 35 - 4.9(x - 3.2)2. a. What are the coordinates of the vertex? b. What are the x-intercepts? c. Where does the parabola intersect the line y = 15?
Write the equation of the circle with center (2.6, -4.5) and radius 3.6.
Draw a right triangle for each problem. Label the sides and angle, then solve to find the unknown measure. a. sin 20( = (/12 b. cos 80( = 25/b c. tan 55( = c + 4 /c d. sin-1 17/30 = D
For each triangle, find the length of the labeled side.a.b. c.
For each triangle, find the measure of the labeled angle.a.b. c.
Draw a pair of intersecting horizontal and vertical lines, and label north, east, south, and west. Sketch the path of a plane flying on a bearing of 30°. a. What is the measure of the angle between the plane's path and the horizontal axis? b. Choose any point along the plane's path. From this
Consider the parametric equations x = t cos 39° y = t sin 39° a. Make a graph with window -1 (x ( 10, -1 (y ( 10, and 0 ( t ( 10. b. Describe the graph. What happens if you change the minimum and maximum values of t? c. Find the measure of the angle between this line and the x-axis. (Trace to a
Consider the parametric equations x = 5t cos 40° y = 5t sin 40° a. Make a graph with window 0 ( x( 5, -2 (y( 5, and 0 ( t( 1. b. Describe the graph and the measure of the angle between the graph and the x-axis. c. What is the relationship between this angle and the parametric equations? d. What
A plane is flying at 100 mi/h on a bearing of 60°. a. Draw a diagram of the motion. Draw a segment perpendicular to the x-axis to create a right triangle. Write equations for x and y, in terms of t, to model the motion. b. What range of t is required to display 500 mi of plane travel? (Assume t
For the plane in Exercise 9 to land in Albuquerque, it must head a bit north. Let A represent the measure of the angle north of west. a. For the plane to fly directly west, the northward component of the plane's motion and the wind's southward component must sum to zero. At what angle must the
A plane is flying on a bearing of 310° at a speed of 320 mi/h. The wind is blowing directly from the east at a speed of 32 mi/h. a. Make a compass rose and vector to indicate the plane's motion. b. Write equations that model the plane's motion without the wind. c. Make a compass rose and vector to
Angelina wants to travel directly across the Wyde River, which is 2 mi wide in this stretch. Her boat can move at a speed of 4 mi/h. The river current flows south at 3 mi/h. At what angle upstream should she aim the boat so that she ends up going straight across?
Two forces are pulling on an object. Force A has magnitude 50 newtons (N) and pulls at an angle of 40°, and Force B has magnitude 90 N and pulls at an angle of 140°, as shown. (Note that the lengths of the vectors show their magnitude.)a. Find the x- and y-components of Force A.b. Find the
A rectangle's length is three times the width. Find the angles, to the nearest degree, at which the diagonals intersect.
Without graphing, determine whether each quadratic equation has no real roots, one real root, or two real roots. If a root is real, indicate whether it is rational or irrational. a. y = 2x2 - 5x - 3 b. y = x2 + 4x - 1 c. y = 3x2 - 3x + 4 d. y = 9x2 - 12x + 4
Consider this system of equations:a. Write the augmented matrix for the system of equations. b. Use row reduction to write the augmented matrix in row-echelon form. Show each step and indicate the operation you use.
Draw a compass rose and vector with magnitude v to find the bearing of each direction. a. 14° south of east b. 14° east of south c. 14° south of west d. 14° north of west
Draw a compass rose and vector with magnitude v for each bearing. Find the angle made with the x-axis. a. 147° b. 204° c. 74° d. 314°
Give the sign of each component vector for the bearings in Exercise 3. For example, for a bearing of 290°, x is negative and y is positive.
A river is 0.3 km wide and flows south at a rate of 7 km/h. You start your trip on the river's west bank, 0.5 km north of the dock, as shown in the diagram at right.a. If the dock is at the origin, (0, 0), what are the coordinates of the boat's starting location?b. Write an equation for x in terms
A pilot wants to fly from Toledo, Ohio, to Chicago, Illinois, which lies 280 mi directly west. Her plane can fly at 120 mi/h. She ignores the wind and heads directly west. However, there is a 25 mi/h wind blowing from the south.a. Write the equation that describes the effect of the wind.b. Write
Fred rows his small boat directly across a river, which is 4 mi wide. There is a 5 mi/h current. When he reaches the opposite shore, Fred finds that he has landed at a point 2 mi downstream.a. Write the equation that describes the effect of the river current.b. If Fred's boat can go s mi/h, what
A plane takes off from Orlando, Florida, heading 975 mi due north toward Cleveland, Ohio. The plane flies at 250 mi/h, and there is a 25 mi/h wind blowing from the west.a. Where is the plane after it has traveled 975 mi north?b. How far did the plane actually travel?c. How fast did the plane
A plane is headed from Memphis, Tennessee, to Albuquerque, New Mexico, 1000 mi due west. The plane flies at 250 mi/h, and the pilot encounters a 20 mi/h wind blowing from the northwest. (That means the direction of the wind makes a 45° angle below the x-axis.)a. Write an equation modeling the
A projectile's motion is described by the equationsx = -50t cos 30° + 400 y = -81t2 + 50t sin 30° + 700 a. Is this projectile motion occurring on Earth, the Moon, or Mars? What are the units used in the problem? (Hint: Look for the value of g.) b. What is the initial position? Include units
A golf ball rolls off the top step of a flight of 14 stairs with a horizontal velocity of 5 ft/s. The stairs are each 8 in. high and 8 in. wide. On which step does the ball first bounce?
A golfer swings a 7-iron golf club with a loft of 38° and an initial velocity of 122 ft/s on level ground. a. Write parametric equations to simulate this golf shot. b. How far away does the ball first hit ground? c. How far away would the ball land if a golfer chose a 9-iron golf club with a loft
The tip of a metronome travels on a path modeled by the parametric equations x = 0.7 sin t and y = (1 - (0.7 sint)2.a. Sketch the graph when 0° ( t ( 360°. Describe the motion.b. Eliminate the parameter, t, and write a single equation using only x and y. Sketch this graph.c. Compare the two
Two forces act simultaneously on a ball positioned at (4, 3). The first force imparts a velocity of 2.3 m/s to the east, and the second force imparts a velocity of 3.8 m/s to the north.a. Enter parametric equations into your calculator to simulate the resulting movement of the ball. Sketch your
Find the height, width, area, and distance specified.a. Total height of the treeb. Width of the lake c. Area of the triangle d. Distance from the boat to the lighthouse
Find a polynomial equation of least degree with integer coefficients that has roots -3 and
Give parametric equations and a single equation using only x and y for the ellipse at right.
Find the position at the time given of a projectile in motion described by the equations x = -50t cos 30° + 40 y = -81t2 + 50t sin 30° + 60 a. 0 seconds b. 1 second c. 2 seconds d. 4 seconds
A ball rolls off the edge of a 12 m tall cliff at a velocity of 2 m/s. a. Write parametric equations to simulate this motion. b. What equation can you solve to determine when the ball hits the ground? c. When and where does the ball hit the ground? d. Describe a graphing window that you can use to
Consider the scenario in Exercise 3. When and where will the ball hit the ground if the motion occurred a. On the Moon? b. On Mars?
These parametric equations model projectile motion of an object: x = 6t cos 52° y = -4.9t2 + 6t sin 52° + 2 a. Name a graphing window and a range of t-values that allow you to simulate the motion. b. Describe a scenario for this projectile motion. Include a description of every variable and
A ball rolls off a 3 ft high table and lands at a point 1.8 ft away from the table. a. How long did it take for the ball to hit the floor? Give your answer to the nearest hundredth of a second. b. How fast was the ball traveling when it left the table? Give your answer to the nearest hundredth, and
An archer aims at a target 70 m away with diameter 1.22 m. The bull's-eye is 1.3 m above the ground. She holds her bow level at a height of 1.2 m and shoots an arrow with an initial velocity of 83 m/s.a. What equations model this motion? b. Will she hit the target? If not, by how much will she miss
By how much does the ball in Example B clear a 10 ft fence that is 365 ft away if the wind is blowing directly from the fence toward Carolina at 8 mi/h?
Gonzo, the human cannonball, is fired out of a cannon 10 ft above the ground at a speed of 40 ft/s. The cannon is tilted at an angle of 60°. His net hangs 5 ft above the ground. Where does his net need to be positioned so that he will land safely?
Find the length of side
One way to calculate the distance between Earth and a nearby star is to measure the angle between the star and the ecliptic (the plane of Earth's orbit) at 6-month intervals. A star is measured at a 42.13204° angle. Six months later, the angle is 42.13226°. The diameter of Earth's orbit is
When light travels from one transparent medium into another, the rays bend, or refract. Snell's Law of Refraction states thatWhere (1 is the angle of incidence, (2 is the angle of refraction, and n1 and n2 are the indexes of refraction for the two mediums, as shown in the diagram. a. Find the angle
Use the quadratic formula to solve each equation. a. 2x2 - 8x + 5 = 0 b. 3x2 + 4x - 2 = 7
Draw a compass rose and show each vector. Find the x- and y-components of each, and show the angle with the x-axis used to find the components. Indicate the direction of the components with the proper sign. a. 12 units on a bearing of 168° b. 16 units on a bearing of 221°
Give the magnitude and bearing for each vector. a. X-component: -9.1 units Y-component: 4.1 units b. X-component: 16.6 units Y-component: 14.4 units
The value of a building depreciates at a rate of 6% per year. When new, the building is worth $36,500. a. How much is the building worth after 5 years 3 months? b. To the nearest month, when will the building be worth less than $10,000?
Find the volume of the greenhouse at right. Round your answer to the nearest square foot.
Assume (PQR is an acute triangle. Find the measure of (P.
In (XYZ, at right, (Z is obtuse. Find the measures of (X and (Z.
Find the length of in Exercise 3.
Find all of the unknown angle measures and side lengths.a.b.
The Daredevil Cliffs rise vertically from the beach. The beach slopes gently down to the water at an angle of 3° from the horizontal. Scott lies at the water's edge, 50 ft from the base of the cliff, and determines that his line of vision to the top of the cliff makes a 70° angle with the
In an isosceles triangle, one of the base angles measures 42°. The length of each leg is 8.2 cm. a. Find the length of the base. b. Even though you are given one angle and two sides not including the angle, this is not an ambiguous case. Why not?
Venus is 67 million mi from the Sun. Earth is 93 million mi from the Sun. Gayle measures the angle between the Sun and Venus as 14°. At that moment in time, how far is Venus from Earth?
The SS Minnow is lost at sea in a deep fog. Moving on a bearing of 107°, the skipper sees a light at a bearing of 60°. The same light reappears through the fog after the skipper has sailed 1.5 km on his initial course. The second sighting of the light is at a bearing of 34°. How far is
Find the length of
Triangulation is used to locate airplanes, boats, or vehicles that transmit radio signals. The distances to the vehicle are found and the directions calculated by measuring the strength of the signal at three fixed receiving locations. Receiver B is 18 km from Receiver A at a bearing of 122°.

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