Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus
39PEFor Exercises 39-50, solve the system by using the inverse of the coefficient matrix. (See Example 7) 5x3y=2010x7y=50 SeeExercise20forA1.41PE42PE43PEFor Exercises 39-50, solve the system by using the inverse of the coefficient matrix. (See Example 7) xy+z=4x+y2z=12x+z=5 SeeExercise26forA1.45PE46PEFor Exercises 39-50, solve the system by using the inverse of the coefficient matrix. (See Example 7) r2s+t=2r+4s+t=32r2st=148PEFor Exercises 39-50, solve the system by using the inverse of the coefficient matrix. (See Example 7) w2x+5y=3x+2y=1xy=12wx+7y+z=5 SeeExercise33forA1.For Exercises 39-50, solve the system by using the inverse of the coefficient matrix. (See Example 7) w+3x3y=8x2z=42w4x+y+2z=6x+y=0 SeeExercise34forA1.For Exercises 51-58, determine whether the statement is true or false. If a statement is false, explain why. A 32 matrix has a multiplicative inverse.For Exercises 51-58, determine whether the statement is true or false. If a statement is false, explain why. A 23 matrix has a multiplicative inverse.For Exercises 51-58, determine whether the statement is true or false. If a statement is false, explain why. Every square matrix has an inverse.54PE55PEFor Exercises 51-58, determine whether the statement is true or false. If a statement is false, explain why. The matrix abab is invertible.57PEFor Exercises 51-58, determine whether the statement is true or false. If a statement is false, explain why. The inverse of the matrix l4 is itself.Find a 22 matrix that is its own inverse. Answers will vary.Given an invertible 22 matrix A and the nonzero real number k, find the inverse of kA in terms of A1.Given A=3256 , a. Find A1 . b. Find A11 .Given A=3254 , a. Find B1 . b. Find B11 .Given A=a00b, where a and b are nonzero real numbers, find A164PE65PEGiven a A=abcd matrix, for what conditions on a,b,c , and d will the matrix not have an inverse?Given A=abcd , perform row operations on the matrix AIn to find A1 . This confirms the formula for the inverse of an invertible 22 matrix. (See page 880).Show by counterexample that AB1A1B1 . That is, find two matrices A and B for which AB1A1B1 .Physicists know that if each edge of a thin conducting plate is kept at a constant temperature, then the temperature at the interior points is the mean (average) of the four surrounding points equidistant from the interior point. Use this principle in Exercise 69 to find the temperature at points x1,x2,x3 , and x4 . Hint: Set up four linear equations to represent the temperature at points x1,x2,x3 , and x4 . Then solve the system. For example, one equation would be: x1=1436+32+x2+x3For Exercises 70-71, use a graphing utility to find the inverse of the given matrix. Round the elements in the inverse to 2 decimal places. A=0.040.130.080.430.190.330.060.840.010.080.110.460.371.420.030.5271PEFor Exercises 72-73, use a graphing calculator and the inverse of the coefficient matrix to find the solution to the given system. Round to 2 decimal places. log2x+7y+z=4.1e2x3yz=3.7ln10x+y2.2z=7.2For Exercises 72-73, use a graphing calculator and the inverse of the coefficient matrix to find the solution to the given system. Round to 2 decimal places. 11x+yln5z=52.37xy+e3z=27.5x+log81yz=69.8Evaluate the determinant. a. 312102 b. 90402SP3SPEvaluate the determinant by expanding cofactors about the elements in the third column. 249512116Use A to determine if A is invertible. A=3210403120053109Solve the system by using Cramer’s rule. 3x4y=95x+6y=2Solve the system by using Cramer's rule. 5x+3y3z=143z4y+z=2x+7y+z=6Solve the system by using Cramer's rule if possible. Otherwise, use a different method. x+4y=23x+12y=4Associated with every square matrix A is a real number denoted by A called the of A .For a 22 matrix A=abcd,A=abcd= .Given A=aij , the of the element atj is the determinant obtained by deleting the ith row and jth column.4PEThe determinant of a 33 matrix A=a1b1c1a2b2c2a3b3c3 is given by A=a1a2+a3Suppose that the given system has one solution. a1x+b1y=c1a2x+b2y=c2 Cramer’s rule gives the solution as x= and y= , where D=,Dx= , and Dy= .For Exercises 7-16, evaluate the determinant of the matrix. (See Example 1) A=3265For Exercises 7-16, evaluate the determinant of the matrix. (See Example 1) B=712149PE10PEFor Exercises 7-16, evaluate the determinant of the matrix. (See Example 1) E=304012PE13PEFor Exercises 7-16, evaluate the determinant of the matrix. (See Example 1) H=y164y15PE16PEFor Exercise 17-22, refer to the matrix A=aij=61184253710 a. Find the minor of the given element (See Example 2) b. Find the cofactor of the given element. q.a12For Exercise 17-22, refer to the matrix A=aij=61184253710 . a. Find the minor of the given element (See Example 2) b. Find the cofactor of the given element. r.a23For Exercise 17-22, refer to the matrix A=aij=61184253710 . a. Find the minor of the given element (See Example 2) b. Find the cofactor of the given element. s.a31For Exercise 17-22, refer to the matrix A=aij=61184253710 . a. Find the minor of the given element (See Example 2) b. Find the cofactor of the given element. t.a1321PE22PE23PE24PE25PE26PE27PE28PE29PE30PEFor Exercises 23-32, evaluate the determinant of the matrix and state whether the matrix is invertible. (See Examples 3-5) T=3814240511010523For Exercises 23-32, evaluate the determinant of the matrix and state whether the matrix is invertible. (See Examples 3-5) W=2524003148011205For Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) 2x+10y=113x5y=6For Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) 5x8y=34x+7y=13For Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) 10x+4y=76x=7y+236PE37PEFor Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) 5x+y=7x+44x=10y839PEFor Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) x=4y+53x4=12yFor Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) 11x3z=12y+9z=64x+5y=9For Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) 2x+6y=95y+7z=14x3z=843PEFor Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) 5x6y+8z=12x+y4z=53x4yz=245PEFor Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) x+y3z=43x3y+9z=122x+2y6z=8For Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) x2y+3z=15x7y+3z=1x5z=2For Exercises 33-48, solve the system if possible by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. (See Examples 6-8) x+3y5z=102x4y+8z=14x+y3z=5For Exercises 49-50, use Cramer's rule to solve for the indicated variable. x1+2x2+3x34x4=35x2+x4=9x1+4x3=15x32x4=8Solveforx2For Exercises 49-50, use Cramer's rule to solve for the indicated variable. 2x1x2+x3+3x4=10x1+5x3=42x2+x4=14x3+2x4=7Solveforx3Determinants can be used to determine whether three points are collinear (lie on the same line). Given the ordered pairs x1,y1,x2,y2 , and x3,y3 , the points are collinear if the determinant to the right equals zero. For Exercises 51-54, determine if the points are collinear. x1y11x2y21x3y31 3,6,6,10,3,252PEDeterminants can be used to determine whether three points are collinear (lie on the same line). Given the ordered pairs x1,y1,x2,y2 , and x3,y3 , the points are collinear if the determinant to the right equals zero. For Exercises 51-54, determine if the points are collinear x1y11x2y21x3y31 4,3,5,7,8,1454PE55PE56PEFor Exercises 57-58, use the formula at the right to find the area of a triangle with vertices x1,y1,x2,y2 , and x3,y3 . Choose the + or sign so that the value of the area is positive. Area=12x1y11x2y21x3y31 1,0,7,2,4,558PE59PE60PEGiven a square matrix A . elementary row operations (or column operations) performed on A affect the value of A , in the following ways: • Interchanging any two rows (or columns) of A will change the sign of A • Multiplying a row (or column) of A by a constant real number k multiplies A by k . • Adding a multiple of a row (or column) of A to another row (or column) of A does not change the value of A . For Exercises 59-64, demonstrate these three properties Given A=1341andB=2641 , a. Evaluate A . b. Evaluate B c. How are A and B related and how are A and B related?62PE63PE64PEGiven a square matrix A , if either of the following conditions are true, then A=0 . • A row (or column) of A consists entirely of zeros. • One row (or column) is a constant multiple of another row (or column). For Exercises 65-68, demonstrate these two properties Given A=35610 , find A .66PE67PE68PEEvaluate I2 .70PE71PEEvaluate x00x .73PE74PE75PE76PEThe determinant of a square matrix can be computed by expanding the cofactors of the elements in any row or column. How would you choose which row or column?78PE79PEFor Exercises 79-82, use a graphing utility to evaluate the determinant of the matrix. Round to the nearest whole unit. 8.92.33.81.70.94.62.710.114.981PE82PEFor Exercises 1-4, solve the system of equations using a. The substitution method or the addition method (see Sections 8.1 and 8.2). b. Gaussian elimination (see Section 9.1). c. Gauss-Jordan elimination (see Section 9.1). d. The inverse of the coefficient matrix (see Section 9.4). e. Cramer's rule (see Section 9.5). x=3y103x7y=22For Exercises 1-4, solve the system of equations using a. The substitution method or the addition method (see Sections 8.1 and 8.2). b. Gaussian elimination (see Section 9.1). c. Gauss-Jordan elimination (see Section 9.1). d. The inverse of the coefficient matrix (see Section 9.4). e. Cramer's rule (see Section 9.5). 2x=28yx+10y=5For Exercises 1-4, solve the system of equations using a. The substitution method or the addition method (see Sections 8.1 and 8.2). b. Gaussian elimination (see Section 9.1). c. Gauss-Jordan elimination (see Section 9.1). d. The inverse of the coefficient matrix (see Section 9.4). e. Cramer's rule (see Section 9.5). x+2yz=02x+z=42xy+2z=5For Exercises 1-4, solve the system of equations using a. The substitution method or the addition method (see Sections 8.1 and 8.2). b. Gaussian elimination (see Section 9.1). c. Gauss-Jordan elimination (see Section 9.1). d. The inverse of the coefficient matrix (see Section 9.4). e. Cramer's rule (see Section 9.5). x+4y+2z=102y+z=4x+y=2For Exercises 5-8 a. Evaluate the determinant of the coefficient matrix. b. Based on the value of the determinant from part (a), can an inverse matrix or Cramer's rule be used to solve the system? c. Solve the system using an appropriate method. 1.5x2y=33x+4y=12For Exercises 5-8 a. Evaluate the determinant of the coefficient matrix. b. Based on the value of the determinant from part (a), can an inverse matrix or Cramer's rule be used to solve the system? c. Solve the system using an appropriate method. 5x2y=1x0.4y=4For Exercises 5-8 a. Evaluate the determinant of the coefficient matrix. b. Based on the value of the determinant from part (a), can an inverse matrix or Cramer's rule be used to solve the system? c. Solve the system using an appropriate method. x3y+7z=12x+5y11z=3x5y+13z=1For Exercises 5-8 a. Evaluate the determinant of the coefficient matrix. b. Based on the value of the determinant from part (a), can an inverse matrix or Cramer's rule be used to solve the system? c. Solve the system using an appropriate method. x2y+3z=72x+y=1xz=3How are the graphs of these ellipses similar and how are they different? x2100+y2225=1 x12100+y+72225=1For Exercises 2-5, an equation of an ellipse is given. a. Identify the center. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Determine the eccentricity. f. Graph the ellipse. x2289+y264=1For Exercises 2-5, an equation of an ellipse is given. a. Identify the center. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Determine the eccentricity. f. Graph the ellipse. 15x2+9y2=135For Exercises 2-5, an equation of an ellipse is given. a. Identify the center. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Determine the eccentricity. f. Graph the ellipse. x+129+y4225=1For Exercises 2-5, an equation of an ellipse is given. a. Identify the center. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Determine the eccentricity. f. Graph the ellipse. x1216+y229=1For Exercises 6-7, a. Write the equation of the ellipse in standard form. b. Identify the center, vertices, endpoints of the minor axis, and foci. 5x2+8y2+40x16y+48=0For Exercises 6-7, a. Write the equation of the ellipse in standard form. b. Identify the center, vertices, endpoints of the minor axis, and foci. 100x2+64y2100x1575=0For Exercises 8-10, write the standard form of an equation of the ellipse subject to the given conditions. Vertices: 4,0 and 4,0; Foci: 11,0 and 11,09REFor Exercises 8-10, write the standard form of an equation of the ellipse subject to the given conditions. Minor axis parallel to x-axis: Center:2,4; Length of major axis: 20 units; Length of minor axis: 12 unitsJupiter orbits the Sun in an elliptical path with the Sun at one focus. At perihelion, Jupiter is closest to the Sun at 7.4052108km . If the eccentricity of the orbit is 0.0489 , determine the distance at aphelion (farthest point between Jupiter and the Sun). Round to the nearest million km.A bridge over a gorge is supported by an arch in the shape of a semiellipse. The length of the bridge is 400ft and the maximum height is 100ft . Find the height of the arch 50ft from the center. Round to the nearest foot.14REFor Exercises 15-16, an equation of a hyperbola is given. Determine whether the transverse axis is horizontal or vertical. x211y216=1For Exercises 15-16, an equation of a hyperbola is given. Determine whether the transverse axis is horizontal or vertical. x211+y216=1For Exercises 17-20, an equation of a hyperbola is given. a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. x29y24=1For Exercises 17-20, an equation of a hyperbola is given. a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. 3x2+2y2=18For Exercises 17-20, an equation of a hyperbola is given. a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. x+3216+y229=1For Exercises 17-20, an equation of a hyperbola is given. a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. x124y+52=1For Exercises 21-22, a. Write the equation of the hyperbola in standard form. b. Identify the center, vertices, and foci. 11x27y2+44x56y145=0For Exercises 21-22, a. Write the equation of the hyperbola in standard form. b. Identify the center, vertices, and foci. 9x2+y2+2y8=0For Exercises 23-25, write the standard form of an equation of a hyperbola subject to the given conditions. Vertices: 4,0,4,0; Foci: 5,0,5,0For Exercises 23-25, write the standard form of an equation of a hyperbola subject to the given conditions. Vertices: 0,13,0,11; Slope of asymptotes: 125For Exercises 23-25, write the standard form of an equation of a hyperbola subject to the given conditions. Corners of reference rectangle: 19,15,19,1,15,1,15,15; Horizontal transverse axisSuppose that one hyperbola has an eccentricity of 4140 and a second hyperbola has an eccentricity of 419 . For which hyperbola do the branches open "wider"?Solve the system of equations. 6x22y2=122x2+y2=11Suppose that two people standing 2mi10,560ft apart at points A and B hear a car backfire. The time difference between the sound heard at point A and the sound heard at point B is 8sec. If sound travels 1100ft/sec, find an equation of the hyperbola (with foci at A and B ) on which the point of origination of the sound must lie.A 120-ft flight control tower in the shape of a hyperboloid has hyperbolic cross sections perpendicular to the ground. Placing the origin at the bottom and center of the tower, the center of a hyperbolic cross section is 0,30 with one focus at 1510,30 and one vertex at 15,30 . All units are in feet. a. Write an equation of a hyperbolic cross section through the origin. Assume that there are no restrictions on x or y . b. Determine the diameter of the tower at the base. Round to the nearest foot. c. Determine the diameter of the tower at the top. Round to the nearest foot.For Exercises 30-33, an equation of a parabola is given in standard form. a. Determine the value of p . b. Identify the vertex. c. Identify the focus. d. Determine the focal diameter. e. Determine the endpoints of the latus rectum. f. Write an equation of the directrix. g. Write an equation for the axis of symmetry. h. Graph the parabola. x2=2yFor Exercises 30-33, an equation of a parabola is given in standard form. a. Determine the value of p . b. Identify the vertex. c. Identify the focus. d. Determine the focal diameter. e. Determine the endpoints of the latus rectum. f. Write an equation of the directrix. g. Write an equation for the axis of symmetry. h. Graph the parabola. 110y2=2xFor Exercises 30-33, an equation of a parabola is given in standard form. a. Determine the value of p . b. Identify the vertex. c. Identify the focus. d. Determine the focal diameter. e. Determine the endpoints of the latus rectum. f. Write an equation of the directrix. g. Write an equation for the axis of symmetry. h. Graph the parabola. y+22=4x333RE34REFor Exercises 34-35, an equation of a parabola is given. a. Write the equation of the parabola in standard form. b. Identify the vertex, focus, and directrix. 4y212y+56x+177=0For Exercises 36-37, fill in the blank. Let p represent the focal length (distance between the vertex and focus). If the directrix of a parabola is given by y=3 and the focus is 4,1, then the vertex is given by the ordered pair , and the value of p is .For Exercises 36-37, fill in the blank. Let p represent the focal length (distance between the vertex and focus). If the vertex of a parabola is 10,4 and the focus is 2,4, then the directrix is given by the equation , and the value of p is .For Exercises 38-40, determine the standard form of the parabola subject to the given conditions. Vertex: 3,2; Directrix: x=5For Exercises 38-40, determine the standard form of the parabola subject to the given conditions. Vertex: 4,7; Focus: 4,140RESolve the system of equations. y+22=3x5xy=742RETwo large airship hangars were built in Orly, France, in the early 1900s but were destroyed in World War II by American aircraft. The hangars were 175m in length, 90m wide, and 60m high and were formed by a series of parabolic arches. a. Set up a coordinate system with 0,0 at the vertex of one of the arches and write an equation of the parabola. b. What is the focal length of an arch?44REFor Exercises 44-45, the given equation represents a conic section (nondegenerative case). Identify the type of conic section. a. 2x2+xy+6y2+4x12y10=0 b. 2x2+10x=y2+8y6 c. x2+4y2=4xyx+6y+346RE47REFor Exercises 46-48, assume that the x- and y-axes are rotated through angle about the origin to form the x- and y-axes . Given =45, write the equation xy+10=0 in xy-coordinates .For Exercises 49-51, an equation of a conic section (nondegenerative case) is given. a. Identify the type of conic section. b. Determine an acute angle of rotation to eliminate the xy term. c. Use a rotation of axes to eliminate the xy term in the equation. d. Sketch the graph. 47x2+343xy+13y264=050RE51REFor Exercises 52-53, the equation represents a conic section (nondegenerative case). a. Identify the type of conic section. b. Graph the equation on a graphing utility. 4x2+3xy2y2+8x12y10=053RE54RE55REFor Exercises 54-59, a. Identify the type of conic represented by the equation. b. Give the eccentricity and write an equation for the directrix in rectangular coordinates. c. Give the coordinates of the vertex or vertices in polar coordinates. d. Sketch the graph. r=223sinFor Exercises 54-59, a. Identify the type of conic represented by the equation. b. Give the eccentricity and write an equation for the directrix in rectangular coordinates. c. Give the coordinates of the vertex or vertices in polar coordinates. d. Sketch the graph. r=11+2cos58RE59RE60RE61RE62RE63RE64REx=4t2 and y=2t for 2t2 Sketch the plane curve by plotting points. Indicate the orientation of the curve.66RE67RE68RE69RE70REFor each given curve, the parameter t represents the time over which an object traverses the curve. Describe the curves and discuss the motion along the curves. C1:x=2cos and y=4sin,0 C2:x=2sin and y=4cos,072RESuppose that the origin on a computer screen is at the lower left comer of the screen. A target moves along a straight line at a constant speed from point A52,410 to point B412,140 in 3sec . a. Write parametric equations to represent the target's path as a function of the time t (in sec) after the target leaves its initial position. All distances are in pixels. b. Where is the target located 0.8sec after motion starts? c. Suppose that a bullet is fired from a gun at a position of 212,150 . Suppose the bullet travels with uniform speed where both the horizontal component of velocity and vertical component of velocity is 40 pixels per second in the positive direction. Write parametric equations to represent the path of the bullet. d. If the bullet leaves the gun at the same time that the target begins its motion, will the bullet hit the target? If so, when and where?A stunt man drives a car at a speed of 25m/sec off a 10-m cliff. The road leading to the edge of the cliff is inclined upward at an angle of 16 . Choose a coordinate system with the origin at the base of the cliff directly under the point where the car leaves the edge. a. Write parametric equations defining the path of the car. b. How long is the car in the air? Round to the nearest tenth of a second. c. How far from the base of the cliff will the car land? Round to the nearest foot.Suppose that a baseball is thrown at an angle of 30 with an initial speed of 88ft/sec from an initial height of 3ft . Choose a coordinate system with the origin at ground level directly under the point of release. a. Write parametric equations defining the path of the ball. b. When will the ball reach its highest point? c. Determine the coordinates of the ball at its highest point. Give the exact values and the coordinates rounded to the nearest tenth of a foot. d. If another player catches the ball at a height of 4ft on the way down, how long was the ball in the air? Round to the nearest hundredth of a second. e. How far apart are the two players? Round to the nearest foot. f. Eliminate the parameter and write an equation in rectangular coordinates to represent the path of the ball.How are the graphs of these ellipses similar, and how are they different? x236+y2144=1x2144+y236=12TWrite the standard form of an equation of an ellipse with center h,k and major axis horizontal.4T5T6TWrite the standard form of an equation of a parabola with vertex h,k and vertical axis of symmetry.Write the standard form of an equation of a parabola with vertex h,k and horizontal axis of symmetry.For Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. x229+y216=1For Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. x229y216=1For Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. x2216+y=0For Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. x2+y24x6y+1=0For Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. 9x2+16y2+64y512=0For Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. 9x2+25y2+72x50y731=015TFor Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. y28y8x+40=0The entrance to a tunnel is in the shape of a semiellipse over a 24-ft by 8-ft rectangular opening. The height at the center of the opening is 14ft . a. Determine the height of the opening at a point 6ft from the edge of the tunnel. Round to 1 decimal place. b. Can a 10-ft high truck pass through the opening if the truck's outer wheel passes at a point 3ft from the edge of the tunnel? c. Set up a coordinate system with the origin placed on the ground in the center of the roadway. Write a function that represents the height of the opening hx (in ft) as a function of the horizontal distance x (in ft) from the center.18TSuppose that two people standing 1.5mi7920ft apart at points A and B hear a car accident. The time difference between the sound heard at point A and the sound heard at point B is 6sec . If sound travels 1100ft/sec, find an equation of the hyperbola (with foci at A and B ) at which the accident occurred.20T21T22TWrite the standard form of an equation of an ellipse with vertices 2,3 and 6,3 and foci 3,3 and 5,3 .Write the standard form of an equation of a hyperbola with vertices 4,1 and 4,7 and foci 4,3 and 4,9 .25TIf the major axis of an ellipse is 26 units in length and the minor axis is 10 units in length, determine the eccentricity of the ellipse.27T28TIf the asymptotes of a hyperbola are y=35x+1 and y=35x+1 , identify the center of the hyperbola.Solve the system of equations. 3x24y2=135x2+2y2=13Describe the graph of the equation. x=41y2932T33T34T35TFor Exercises 36-37, an equation of a conic (nondegenerative case) is given. a. Identify the type of conic. b. Determine an acute angle of rotation to eliminate the xy term. Round to the nearest tenth of a degree if necessary. c. Use a rotation of axes to eliminate the xy term in the equation. d. Sketch the graph. 13x210xy+13y2242x242y=037T38T39T40TFor Exercises 41-42, a. Eliminate the parameter. b. Sketch the curve and show the orientation of the curve. x=2t and y=2+3t for 1t2For Exercises 41-42, a. Eliminate the parameter. b. Sketch the curve and show the orientation of the curve. x=23sin and y=3cos43TA pyrotechnic rocket is fired from a platform 2ft high at an angle of 60 from the horizontal with an initial speed of 72ft/sec . Choose a coordinate system with the origin at ground level directly below the launch position. a. Write parametric equations that model the path of the shell as a function of the time t (in sec) after launch. b. Approximate the time required for the shell to hit the ground. Round to the nearest hundredth of a second. c. Approximate the horizontal distance that the shell travels before it hits the ground. Round to the nearest foot. d. When is the shell at its maximum height? Find the exact value and an approximation to the nearest hundredth of a second. e. Determine the maximum height.45T46TUse the parametric mode to graph the curve on a graphing utility. x=sin4t and y=3cost,0t21CREIs 4 a zero of the polynomial fx=3x46x3+5x12?3CRE4CREFor Exercises 4-5, simplify completely. 2x2y1z344x5y22Find the midpoint of the line segment with endpoints 362,147 and 118,24.Given fx=x73, write an equation defining f1x.8CREDetermine the asymptotes of the graph of y=csc2x2For Exercises 10-15, solve the equation or inequality. Write the solution set to the inequalities in interval notation if possible. e3x+2=22For Exercises 10-15, solve the equation or inequality. Write the solution set to the inequalities in interval notation if possible. 234x+2x13+40For Exercises 10-15, solve the equation or inequality. Write the solution set to the inequalities in interval notation if possible. 2192x+113CREFor Exercises 10-15, solve the equation or inequality. Write the solution set to the inequalities in interval notation if possible. log3x+log3x6=3For Exercises 10-15, solve the equation or inequality. Write the solution set to the inequalities in interval notation if possible. e2x8ex+15=016CREGraph y=2sinx4 over one completed period.18CRE19CRE20CRE21CRE22CREFor Exercises 21-24, graph the equations and functions. x2=4y2For Exercises 21-24, graph the equations and functions. fx=12x+2For Exercises 25-26, solve the system. 3x4y+2z=55y3z=127x+2z=1For Exercises 25-26, solve the system. x25y2=44x2+y2=37For Exercises 27-29, refer to the matrices A and B . Simplify each expression A=2416B=3804 2ABFor Exercises 27-29, refer to the matrices A and B . Simplify each expression A=2416B=3804 A1For Exercises 27-29, refer to the matrices A and B . Simplify each expression A=2416B=3804 BUse Cramer's rule to solve for z . 3x4y+z=32x+3y=85y+8z=11Graph the ellipse, and identify the center, vertices, minor axis endpoints, and foci. x29+y24=1Graph the ellipse and identify the center, vertices, minor axis endpoints, and foci. 25x2+9y2=900Graph the ellipse and identify the center, vertices, foci, and endpoints of the minor axis. x+1225+y229=1Repeat Example 4 with 2x2+11y212x+44y+40=0 .Determine the standard form of an equation of an ellipse with vertices 3,3 and 3,3 and foci. 22,3 and 22,3 .A tunnel has vertical sides of 7 ft with a semielliptical top. The width of the tunnel is 10 ft, and the height at the top is 10 ft. a. Write an equation of the semiellipse. For convenience, place the coordinate system with 0,0 at the center of the ellipse. b. To construct the tunnel, an engineer needs to find the location of the foci. How far from the center are the foci?Pluto is 7.376109 km at aphelion (farthest point from the Sun). Use the eccentricity of Pluto's orbit of 0.2488 to find the closest distance between Pluto and the Sun.The circle, the ellipse, the hyperbola, and the parabola are categories of sections.An is a set of points x,y in a plane such that the sum of the distances between x,y and two fixed points called is a constant.The line through the foci intersects an ellipse at two points called .The line segment with endpoints at the vertices of an ellipse is called the axis.The line segment perpendicular to the major axis, with endpoints on the ellipse, and passing through the center of the ellipse, is called the axis.Given x2a2+y2b2=1, where ab0, the ordered pairs representing the vertices are and . The ordered pairs representing the endpoints of the minor axis are and .Given xh2b2+yk2a2=1, where ab0, the ordered pairs representing the endpoints of the vertices are and . The ordered pairs representing the endpoints of the minor axis are and .When referring to the standard form of an equation of an ellipse, the e is defined as e=.For Exercises 9-10, a. Use the distance formula to find the distances d1,d2,d3 and d4 . b. Find the sum d1+d2 . c. Find the sum d3+d4 . d. How do the sums from parts (b) and (c) compare? e. How do the sums of the distances from parts (b) and (c) relate to the length of the major axis?