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All Textbook Solutions for Precalculus Enhanced with Graphing Utilities

In Problems 13-24, analyze each equation and graph it. r= 9 36cos In Problems 13-24, analyze each equation and graph it. r= 12 4+8sin In Problems 13-24, analyze each equation and graph it. r= 8 2sin In Problems 13-24, analyze each equation and graph it. r= 8 2+4cos In Problems 13-24, analyze each equation and graph it. r( 32sin )=6 In Problems 13-24, analyze each equation and graph it. r( 2cos )=2 23AYU 24AYU In Problems 25-36, convert each polar equation to a rectangular equation. r= 1 1+cos In Problems 25-36, convert each polar equation to a rectangular equation. r= 3 1sin In Problems 25-36, convert each polar equation to a rectangular equation. r= 8 4+3sin In Problems 25-36, convert each polar equation to a rectangular equation. r= 10 5+4cos In Problems 25-36, convert each polar equation to a rectangular equation. r= 9 36cos In Problems 25-36, convert each polar equation to a rectangular equation. r= 12 4+8sin In Problems 25-36, convert each polar equation to a rectangular equation. r= 8 2sin In Problems 25-36, convert each polar equation to a rectangular equation. r= 8 2+4cos In Problems 25-36, convert each polar equation to a rectangular equation. r( 32sin )=6 In Problems 25-36, convert each polar equation to a rectangular equation. r( 2cos )=2 In Problems 25-36, convert each polar equation to a rectangular equation. r= 6sec 2sec1 In Problems 25-36, convert each polar equation to a rectangular equation. r= 3csc csc1 37AYU 38AYU 39AYU 40AYU 41AYU 42AYU 43AYU 44AYU Derive equation (d) in Table 5: r= ep 1esin Orbit of Mercury The planet Mercury travels around the Sun in an elliptical orbit given approximately by r= ( 3.442 ) 10 7 10.206cos where r is measured in miles and the Sun is at the pole. Find the distance from Mercury to the Sun at aphelion (greatest distance from the Sun) and at perihelion (shortest distance from the Sun). See the figure. Use the aphelion and perihelion to graph the orbit of Mercury using a graphing utility.The function f( x )=3sin( 4x ) has amplitude _______ and period _______. (pp. 412-414)Let x=f( t ) and y=g( t ) , where f and g are two functions whose common domain is some interval I . The collection of points defined by ( x,y )=( f( t ),g( t ) ) , is called a(n) ______ ______. The variable t is called a(n) _______.The parametric equations x=2sint , y=3cost define a(n) _______. a. circle b. ellipse c. hyperbola d. parabola 4AYU True or False Parametric equations defining a curve are unique.True or False Curves defined using parametric equations have an orientation.In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=3t+2 , y=t+1 ; 0t4 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=t3 , y=2t+4 ; 0t2 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=t+2 , y= t ; t0 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= 2t , y=4t ; t0 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= t 2 +4 , y= t 2 4 ; t In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= t +4 , y= t 4 ; t0 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=3 t 2 , y=t+1 ; t In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=2t4 , y=4 t 2 ; t In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=2 e t , y=1+ e t ; t0 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= e t , y= e t ; t0 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= t , y= t 3/2 ; t0 t0 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= t 3/2 +1 , y= t ; t0 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=2cost , y=3sint ; 0t2 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=2cost , y=3sint ; 0t In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=2cost , y=3sint ; t0 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=2cost , y=sint ; 0t 2 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=sect , y=tant ; 0t 4 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x=csct , y=cott ; 4 t 2 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= sin 2 t , y= cos 2 t ; 0t2 In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility. x= t 2 , y=lnt ; t0 y=4x1 y=8x+3 y= x 2 +1 y= x 3 y= x 3 y= x 4 +1 x= y 3/2 x= y In Problems 35-38, find parametric equations that define the curve shown.In Problems 35-38, find parametric equations that define the curve shown.In Problems 35-38, find parametric equations that define the curve shown.In Problems 35-38, find parametric equations that define the curve shown.In Problems 39-42, find parametric equations for an object that moves along the ellipse x 2 4 + y 2 9 =1 with the motion described. The motion begins at ( 2,0 ) , is clockwise, and requires 2 seconds for a complete revolution.In Problems 39-42, find parametric equations for an object that moves along the ellipse x 2 4 + y 2 9 =1 with the motion described. The motion begins at ( 0,3 ) , is counterclockwise, and requires 1 seconds for a complete revolution.In Problems 39-42, find parametric equations for an object that moves along the ellipse x 2 4 + y 2 9 =1 with the motion described. The motion begins at ( 0,3 ) , is clockwise, and requires 1 seconds for a complete revolution.In Problems 39-42, find parametric equations for an object that moves along the ellipse x 2 4 + y 2 9 =1 with the motion described. The motion begins at ( 2,0 ) , is counterclockwise, and requires 3 seconds for a complete revolution.In Problems 43 and 44, the parametric equations of four curves are given. Graph each of them, indicating the orientation. C 1 :x=t,y= t 2 ;4t4 C 2 :x=cost,y=1 sin 2 t;0t C 3 :x= e t ,y= e 2t ;0tln4 C 4 :x= t ,y=t;0t16 In Problems 43 and 44, the parametric equations of four curves are given. Graph each of them, indicating the orientation. C 1 :x=t,y= 1 t 2 ;1t1 C 2 :x=sint,y=cost;0t2 C 3 :x=cost,y=sint;0t2 C 4 :x= 1 t 2 ,y=t;1t1 In Problems 45-48, use a graphing utility to graph the curve defined by the given parametric equations. x=tsint , y=tcost , t0 In Problems 45-48, use a graphing utility to graph the curve defined by the given parametric equations. x=sint+cost , y=sintcost In Problems 45-48, use a graphing utility to graph the curve defined by the given parametric equations. x=4sint2sin( 2t ) y=4cost2cos( 2t )In Problems 45-48, use a graphing utility to graph the curve defined by the given parametric equations. x=4sint+2sin( 2t ) y=4cost+2cos( 2t )Projectile Motion Bob throws a ball straight up with an initial speed of 50 feet per second from a height of 6 feet. a. Find parametric equations that model the motion of the ball as a function of time. b. How long is the ball in the air? c. When is the ball at its maximum height? Determine the maximum height of the ball. d. Simulate the motion of the ball by graphing the equations found in part (a).Projectile Motion Alice throws a ball straight up with an initial speed of 40 feet per second from a height of 5 feet. a. Find parametric equations that model the motion of the ball as a function of time. b. How long is the ball in the air? c. When is the ball at its maximum height? Determine the maximum height of the ball. d. Simulate the motion of the ball by graphing the equations found in part (a).Catching a Train Billâ€™s train leaves at 8:06 AM and accelerates at the rate of 2 meters per second per second. Bill, who can run 5 meters per second, arrives at the train station 5 seconds after the train has left and runs for the train. a. Find parametric equations that model the motions of the train and Bill as a function of time. [Hint: The position s at time t of an object having acceleration a is s= 1 2 a t 2 ]. b. Determine algebraically whether Bill will catch the train. If so, when? c. Simulate the motion of the train and Bill by simultaneously graphing the equations found in part (a).Catching a Bus Jodiâ€™s bus leaves at 5:30 PM and accelerates at the rate of 3 meters per second per second. Jodi, who can run 5 meters per second, arrives at the bus station 2 seconds after the bus has left and runs for the bus. a. Find parametric equations that model the motions of the bus and Jodi as a function of time. [Hint: The position s at time t of an object having acceleration a is s= 1 2 a t 2 .] b. Determine algebraically whether Jodi will catch the bus. If so, when? c. Simulate the motion of the bus and Jodi by simultaneously graphing the equations found in part (a).Projectile Motion Ichiro throws a baseball with an initial speed of 145 feet per second at an angle of 20 to the horizontal. The ball leaves Ichiroâ€™s hand at a height of 5 feet. a. Find parametric equations that model the position of the ball as a function of time. b. How long is the ball in the air? c. Determine the horizontal distance that the ball travels. d. When is the ball at its maximum height? Determine the maximum height of the ball. e. Using a graphing utility, simultaneously graph the equations found in part (a).Projectile Motion Mark Texeira hit a baseball with an initial speed of 125 feet per second at an angle of 40 to the horizontal. The ball was hit at a height of 3 feet off the ground. a. Find parametric equations that model the position of the ball as a function of time. b. How long was the ball in the air? c. Determine the horizontal distance that the ball traveled. d. When was the ball at its maximum height? Determine the maximum height of the ball. e. Using a graphing utility, simultaneously graph the equations found in part (a).Projectile Motion Suppose that Adam hits a golf ball off a cliff 300 meters high with an initial speed of 40 meters per second at an angle of 45 to the horizontal. a. Find parametric equations that model the position of the ball as a function of time. b. How long is the ball in the air? c. Determine the horizontal distance that the ball travels. d. When is the ball at its maximum height? Determine the maximum height of the ball. e. Using a graphing utility, simultaneously graph the equations found in part (a).Projectile Motion Suppose that Karla hits a golf ball off a cliff 300 meters high with an initial speed of 40 meters per second at an angle of 45 to the horizontal on the Moon (gravity on the Moon is one-sixth of that on Earth). a. Find parametric equations that model the position of the ball as a function of time. b. How long is the ball in the air? c. Determine the horizontal distance that the ball travels. d. When is the ball at its maximum height? Determine the maximum height of the ball. e. Using a graphing utility, simultaneously graph the equations found in part (a).Uniform Motion AToyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the intersection when the Impala is 4 miles from the intersection. See the figure. a. Find parametric equations that model the motion of the Camry and Impala. b. Find a formula for the distance between the cars as a function of time. c. Graph the function in part (b) using a graphing utility. d. What is the minimum distance between the cars? When are the cars closest? e. Simulate the motion of the cars by simultaneously graphing the equations found in part (a).Uniform Motion A Cessna (heading south at 120 mph) and a Boeing 747 (heading west at 600 mph) are flying toward the same point at the same altitude. The Cessna is 100 miles from the point where the flight patterns intersect, and the 747 is 550 miles from this intersection point. See the figure. a. Find parametric equations that model the motion of the Cessna and the 747. b. Find a formula for the distance between the planes as a function of time. c. Graph the function in part (b) using a graphing utility. d. What is the minimum distance between the planes? When are the planes closest? e. Simulate the motion of the planes by simultaneously graphing the equations found in part (a).The Green Monster The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of 45 . Use g=32 feet per second2 as the acceleration due to gravity and ignore any air resistance. a. Find parametric equations that model the position of the ball as a function of time. What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet. b. How far is the ball from home plate at its maximum height? Give your answer in feet. c. If the ball is hit straight down the left field wall, will it clear the Green Monster? If it does, by how much does it clear the wall?Projectile Motion The position of a projectile fired with an initial velocity 0 feet per second and at an angle to the horizontal at the end of t seconds is given by the parametric equations x=( 0 cos )t y=( 0 sin )t16 t 2 See the illustration. a. Obtain the rectangular equation of the trajectory and identify the curve. b. Show that the projectile hits the ground ( y=0 ) when t= 1 16 0 sin . c. How far has the projectile traveled (horizontally) when it strikes the ground? In other words, find the range R . d. Find the time t when x=y . Then find the horizontal distance x and the vertical distance y traveled by the projectile in this time. Then compute x 2 + y 2 . This is the distance R , the range, that the projectile travels up a plane inclined at 45 to the horizontal ( x=y ) . See the following illustration. (See also Problem 99 in Section 7.6.)Show that the parametric equations for a line passing through the points ( x 1 , y 1 ) and ( x 2 , y 2 ) are x=( x 2 x 1 )t+ x 1 y=( y 2 y 1 )t+ y 1 , t What is the orientation of this line?Hypocycloid The hypocycloid is a curve defined by the parametric equations x( t )= cos 3 t , y( t )= sin 3 t , 0t2 a. Graph the hypocycloid using a graphing utility. b. Find a rectangular equation of the hypocycloid.In Problem 62, we graphed the hypocycloid. Now graph the rectangular equations of the hypocycloid. Did you obtain a complete graph? If not, experiment until you do.64AYU 1RE 2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE 33RE 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 45RE 46RE 47RE 48RE 49RE 50RE 51RE 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE 1CT 2CT 3CT 4CT 5CT 6CT 7CT 8CT 9CT 10CT 11CT 12CT 13CT 14CT 15CT 16CT 17CT 18CT 19CT 20CT 21CT 22CT 23CT 24CT 25CT 26CT 27CT 28CT 1CR 2CR 3CR 4CR 5CR 6CR 7CR 8CR 9CR 10CR 11CR 12CR 1AYU 2AYU 3AYU 4AYU 5AYU 6AYU 7AYU 8AYU 9AYU 10AYU 11AYU 12AYU 13AYU 14AYU 15AYU 16AYU 17AYU 18AYU 19AYU 20AYU 21AYU 22AYU 23AYU 24AYU 25AYU 26AYU 27AYU 28AYU 29AYU 30AYU 31AYU 32AYU 33AYU 34AYU 35AYU 36AYU 37AYU 38AYU 39AYU 40AYU 41AYU 42AYU 43AYU 44AYU 45AYU 46AYU 47AYU 48AYU 49AYU 50AYU 51AYU 52AYU 53AYU 54AYU 55AYU 56AYU 57AYU 58AYU 59AYU 60AYU 61AYU 62AYU 63AYU 64AYU 65AYU 66AYU 67AYU 68AYU 69AYU 70AYU 71AYU 72AYU 73AYU 74AYU 75AYU 76AYU 77AYU 78AYU 79AYU 80AYU 81AYU 82AYU 83AYU 84AYU An m by n rectangular array of numbers is called a( n ) _____ .The matrix used to represent a system of linear equations is called a( n ) _____ matrix.The notation a 35 refers to the entry in the _____ row and _____ column of a matrix.True or False The matrix [ 1 0 0 3 1 0 | 2 5 0 ] is in row echelon form.In problems 7-18, write the augmented matrix of the given system of equations. { x5y=5 4x+3y=6 In problems 7-18, write the augmented matrix of the given system of equations. { 3x+4y=7 4x2y=5 In problems 7-18, write the augmented matrix of the given system of equations. { 2x+3y6=0 4x6y+2=0 In problems 7-18, write the augmented matrix of the given system of equations. { 9xy=0 3xy4=0 In problems 7-18, write the augmented matrix of the given system of equations. { 0.01x0.03y=0.06 0.13x+0.10y=0.20 In problems 7-18, write the augmented matrix of the given system of equations. { 4 3 x 3 2 y= 3 4 1 4 x+ 1 3 y= 2 3 In problems 7-18, write the augmented matrix of the given system of equations. { xy+z=10 3x+3y=5 x+y+2z=2 In problems 7-18, write the augmented matrix of the given system of equations. { 5xyz=0 x+y=5 2x3z=2 In problems 7-18, write the augmented matrix of the given system of equations. { x+yz=2 3x2y=2 5x+3yz=1 In problems 7-18, write the augmented matrix of the given system of equations. { 2x+3y4z=0 x5z+2=0 x+2y3z=2 In problems 7-18, write the augmented matrix of the given system of equations. { xyz=10 2x+y+2z=1 3x+4y=5 4x5y+z=0 In problems 7-18, write the augmented matrix of the given system of equations. { xy+2zw=5 x+3y4z+2w=2 3xy5zw=1 In Problems 19-26, write the system of equations corresponding to each augmented matrix. Then perform the indicated row operation(s) on the given augmented matrix. [ 1 2 3 5 | 2 5 ] R 2 =2 r 1 + r 2 In Problems 19-26, write the system of equations corresponding to each augmented matrix. Then perform the indicated row operation(s) on the given augmented matrix. [ 1 2 3 5 | 3 4 ] R 2 =2 r 1 + r 2 In Problems 19-26, write the system of equations corresponding to each augmented matrix. Then perform the indicated row operation(s) on the given augmented matrix. [ 1 3 5 3 5 3 4 6 4 | 3 6 6 ] R 2 =3 r 1 + r 2 R 3 =5 r 1 + r 3 In Problems 19-26, write the system of equations corresponding to each augmented matrix. Then perform the indicated row operation(s) on the given augmented matrix. [ 1 4 3 3 5 2 3 3 4 | 5 5 6 ] R 2 =4 r 1 + r 2 R 3 =3 r 1 + r 3

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