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All Textbook Solutions for Precalculus

When applying the law of cosines (or law of sines) how can you minimize round-off error?Suppose that ABC is a right triangle with right angle at vertex C . What is the result when the law of cosines is used to find the length of side c ?If the measures of the three angles in a triangle are known, can you solve for the lengths of the sides?A piston in an engine is attached to a connector rod of length L at the wrist pin at point P . As the piston travels back and forth, the connector cap at point Q travels counterclockwise along a circular path of radius r . If L2r , and a represents PCQ . a. Use the law of cosines to show that the distance x from the wrist pin to the crank circle is given by x=rcos+r2cos2ar2+L2r . b. Find the distance between the wrist pin and crank circle for a truck engine with connector rod of length 5.9in. and crank radius of 1.7in. when a=105. Round to the nearest tenth of an inch, c. Find the maximum and minimum distances that the wrist pin is to the crank circle, using the values from part (b).Use the law of cosines to show that cosAa+cosBb+cosCc=a2+b2+c22abc Use the law of cosines to show that a=bcosC+ccosB 60PE For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. a=20,b=29,B=90 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. b=16.2,c=12,A=113 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. a=16.2,b=18,5,c=28.6 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. a=19,b=26,A=72 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. c=20,b=17,A=90 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. a=14.1,A=28,B=110 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. a=7,b=8.8,A=40 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. b=18,c=26,C=56 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. b=108,B=62,C=54 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. a=12,c=48,B=62 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. a=106,b=58,c=172 For Exercises 1-12, solve ABC subject to the given conditions, if possible. Round the lengths of the sides and the measures of the angles to 2 decimal places, if necessary. a=46,b=61,B=136 Refer to the figure. Find the indicated values to the nearest hundredth of a centimeter or hundredth of a degree. a. Find the lengths of BC,AC,DB,DC,ED,EC, and EA . b. Find the measure of AEC and CAE .Use the right triangles APB and APC to show that for ABC , the length of side a is given by a=bcosC+ccosB.A hiking trail is roughly in the shape of a circle with radius 2000ft . Suppose that a rectangular coordinate system is set up with the origin at the center of the circle. Donna and Eddie start walking from point A2000,0 at 220ft/min . Eddie goes off-trail and walks directly toward the center of the circle and Donna walks along the trail counterclockwise. a. Find the coordinates of each person after 5min . Round the coordinates to the nearest foot b. How far apart are Eddie and Donna after 5min ? Round to the nearest foot. c. If Eddie can no longer walk because he sprains his ankle after foolishly leaving the trail, at what bearing would Donna walk to find him? Round to the nearest tenth of a degree.Suppose the block from Example 1 is initially pulled 6 in. below the equilibrium position. Repeat Example 1 if it takes the block 0.25sec to return to its starting position.Repeat Example 2 with a Ferris wheel 120ft in diameter that completes one revolution in 1.25min .Two spring-mass systems have an initial displacement of 6cm and then oscillate at 0.25Hz . The damping factors are 0.4 and 0.1 , for spring 1 and spring 2, respectively. Write a function to model the displacement for each system t seconds after release.The of an object in simple harmonic motion is the amount of time required for one complete cycle.For an object in simple harmonic motion, the number of cycles per unit time is called the .Given an object in damped harmonic motion, the amplitude (increases/decreases) with time.The term "ultrasound" refers to sound waves of a frequency greater than those detectable by the human ear 20,000Hz Suppose that a medical imaging device produces ultrasound with a period of 5s (micro seconds: note that 5s=0.000005sec ). What is the frequency?The frequency of middle C on a piano is. 264Hz . What is the time required for one complete cycle?If the displacement d of an object moving under simple harmonic motion is maximized at time t=0 , which model would be most convenient? d=asint or d=acost For Exercises 7-14, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase shift e. Least positive value of t for which d=0 d=6cos80t For Exercises 7-14, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase shift e. Least positive value of t for which d=0 d=2cos120t For Exercises 7-14, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase shift e. Least positive value of t for which d=0 d=0.25sin4t For Exercises 7-14, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase shift e. Least positive value of t for which d=0 d=0.125sin2t For Exercises 7-14, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase shift e. Least positive value of t for which d=0 d=cost6 For Exercises 7-14, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase shift e. Least positive value of t for which d=0 d=cost+4 For Exercises 7-14, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase shift e. Least positive value of t for which d=0 d=14sin12t+1 For Exercises 7-14, suppose that an object moves in simple harmonic motion with displacement d (in centimeters) at time t (in seconds). Determine the a. Amplitude b. Period c. Frequency d. Phase shift e. Least positive value of t for which d=0 d=13sin14t+12 For Exercises 15-22, suppose that an object is attached to a horizontal spring subject to the given conditions. Find a model for the displacement d as a function of the time t . (See Example 1) InitialDisplacementdatt=0AmplitudePeriodorFrequency2.5in.2.5in.1.2sec For Exercises 15-22, suppose that an object is attached to a horizontal spring subject to the given conditions. Find a model for the displacement d as a function of the time t . (See Example 1) InitialDisplacementdatt=0AmplitudePeriodorFrequency4.1cm4.1cm2sec 17PE For Exercises 15-22, suppose that an object is attached to a horizontal spring subject to the given conditions. Find a model for the displacement d as a function of the time t . (See Example 1) InitialDisplacementdatt=0AmplitudePeriodorFrequency12m12m4Hz For Exercises 15-22, suppose that an object is attached to a horizontal spring subject to the given conditions. Find a model for the displacement d as a function of the time t . (See Example 1) InitialDisplacementdatt=0AmplitudePeriodorFrequency0cmintiallymovingtotheright8.5cm0.2sec For Exercises 15-22, suppose that an object is attached to a horizontal spring subject to the given conditions. Find a model for the displacement d as a function of the time t . (See Example 1) InitialDisplacementdatt=0AmplitudePeriodorFrequency0cmintiallymovingtotheright15cm0.125sec For Exercises 15-22, suppose that an object is attached to a horizontal spring subject to the given conditions. Find a model for the displacement d as a function of the time t . (See Example 1) InitialDisplacementdatt=0AmplitudePeriodorFrequency0ftintiallymovingtotheleft0.25ft0.1Hz For Exercises 15-22, suppose that an object is attached to a horizontal spring subject to the given conditions. Find a model for the displacement d as a function of the time t . (See Example 1) InitialDisplacementdatt=0AmplitudePeriodorFrequency0mintiallymovingtotheletf1m0.01Hz A block hangs on a spring attached to the ceiling and is pulled down 5in . below its equilibrium position. After release, the block makes one complete up-and-down cycle in 1sec and follows simple harmonic motion. (See Example 1) a. What is the period of motion? b. What is the frequency? c. What is the amplitude? d. Write a function to model the displacement d (in inches) as a function of the time t (in seconds) after release. Assume that a displacement above the equilibrium position is positive. e. Find the displacement of the block and direction of movement at t=0.125sec .The bob on a simple pendulum is pulled to the left 4 in. from its equilibrium position. After release, the pendulum makes one complete back-and-forth cycle in 2sec and follows simple harmonic motion. a. What is the period of motion? b. What is the frequency? c. What is the amplitude? d. Write a function to model the displacement (in inches) of the bob as a function of the time t (in seconds) after release. Assume that a displacement to the right of the equilibrium position is positive. e. Find the position and direction of movement of the bob at t=1.25sec .25PE The blood pressure p for a certain individual follows a pattern of simple harmonic motion during the pumping cycle between heartbeats. The minimum pressure is 80 mmHg (millimeters of mercury) and the maximum pressure is 120 mmHg. The individual's pulse is 60 beats per minute or equivalently 1 beat per second. Write a model representing the blood pressure p at a time t seconds into the cycle. Assume that at t=0 , the blood pressure is 100 mmHg and is initially increasing.A Ferris wheel at a county Fair is 180ft in diameter with its lowest point 2.5ft off the ground. Once all the passengers have been loaded, the wheel makes one full rotation counterclockwise in 1.2min . Suppose that two children are seated at the lowest point on the wheel and are the last passengers to be loaded when the wheel starts. a. Write a model representing the children's horizontal position x (in feet) relative to the center of the Ferris wheel, t minutes after the ride starts. b. Write a model representing the children's height y (in feet) above ground level, t minutes after the ride starts. c. Give the coordinates of the children's position 1min into the ride and describe the location.The 30-ft diameter waterwheel at Deleon Springs State Park is an old sugar mill that turns at a rate of 2rev/min , counterclockwise. Suppose that a coordinate system is chosen with the origin at the bottom of the wheel (at water level). At t=0 , suppose that a point on the wheel is located at 0,0 . a. Write a function that represents the x position (in feet) of the point as a function of the time t in minutes. b. Write a function that represents they position (in feet) of the point as a function of the time t in minutes. c. Give the coordinates of the point at t=40sec and describe the location.Refer to the piston and crankshaft from Exercise 25. The stroke length of the engine is 4 in. and the engine turn at 1800 rpm. The connecting rod cap at point B rotates around the crank circle centered at point C Assume that at t=0 , the piston is at its farthest point from the center of the crank circle. a. How many revolutions per second is the engine turning? b. Write a model representing the horizontal position x (in inches) of the connecting rod cap (point B ) relative to the center of the crank circle (point C ) at time t (in seconds). c. Write a model representing the vertical position y (in inches) of the connecting rod cap relative to the center of the crank circle at time t (in seconds).A laboratory centrifuge is a piece of equipment that spins blood samples at a high speed to separate substances within the blood of greater or lesser density. Suppose that a test tube is placed at a point 10cm to the right of the center of rotation and that the centrifuge spins at 3600rpm counterclockwise, a. How many revolutions does the centrifuge make per second? b. Write a model representing the horizontal positionx (in centimeters) of the test tube relative to the center of the centrifuge at time t (in seconds), c. Write a model representing the vertical position y (in centimeters) of the test tube relative to the center of the centrifuge at time t (in seconds).An alternating current generator generates current with a frequency of 60Hz . Suppose that initially, the current is at its maximum of 5 amperes. If the current varies in simple harmonic motion over time, write a model for the current l (in amperes) as a function of the time t (in seconds).An alternating current generator in the United Kingdom generates current with a frequency of 50Hz . Suppose that initially, the current is at its maximum of 12 amperes. If the current varies in simple harmonic motion over time, write a model for the current l (in amperes) as a function of the time t (in seconds).33PE 34PE The brightness of a "young" star sometimes increases and decreases as a result of regional areas of "hot" and "cold" on the star's surface as well as variations in the density of the star's planet-forming debris, which can obstruct light Suppose that for a particular star, the average magnitude (measure of brightness) is 4.3 with a variation of 0.31 (on the magnitude scale, brighter objects have a smaller magnitude than dimmer objects). Furthermore, the magnitude of a star is initially observed to be 4.61 , and the time between minimum brightness and maximum brightness is 6.4 days. Write a simple harmonic motion model to describe the magnitude M of the star for day t .The magnitude of a star named Delta Cuphea varies from an apparent magnitude of 3.6 to an apparent magnitude of 4.3 with a period of 5.4 days. At t=0 days, the star is at its brightest with a magnitude of 3.6 (on the magnitude scale, brighter objects have a smaller magnitude than dimmer objects). Write a simple harmonic motion model to describe the magnitude M of the star for day t .For Exercises 37-38, the ordered pair t,d gives the displacement d (in centimeters) of an object undergoing simple harmonic motion at time t (in seconds). Suppose that the object has a minimum at 24,32 and next consecutive maximum at 48,54 . a. What is the period? b. What is the frequency? c. What is the amplitude? d. Write a model representing the displacement as a function of time.For Exercises 37-38, the ordered pair t,d gives the displacement d (in centimeters) of an object undergoing simple harmonic motion at time t (in seconds). Suppose that the object has a maximum at 4,16 and then returns to its next equilibrium position at 6,11 . a. What is the period? b. What is the frequency? c. What is the amplitude? d. Write a model representing the displacement as a function of time.Write a function y=ft for simple harmonic motion whose graph has a minimum at 4,6 and next consecutive equilibrium at 2,2 .Write a function y=ft for simple harmonic motion whose graph has a maximum at 3,8 and next consecutive minimum at ,2 .Use a graphing utility to graph the functions of the form d=aacost and comment on the role of the damping factor c0 in the equation. a. d=6e0.1tcost b. d=6e0.3tcost c. d=60.5tcost 42PE 43PE For Exercises 43-46, use a graphing utility to graph the function and bounding curves for t0 . d=12e0.55tsin16t For Exercises 43-46, use a graphing utility to graph the function and bounding curves for t0 . d=8e1.2tcos4t For Exercises 43-46, use a graphing utility to graph the function and bounding curves for t0 . d=4e0.008tcos8t Suppose that a guitar string is plucked such that the center of the string is initially displaced 10mm and then vibrates under damped harmonic motion. The note produced has a frequency of110Hz . The note is no longer audible to a normal human ear once the displacement at the middle of the string is less than 0.1mm . What is the damping factor if the sound is no longer audible after2.5sec ? Round to 2 decimal places.A tuning fork is struck, and the tips of the prongs (tines) are initially displaced 0.4mm from their natural position at rest. The prongs oscillate at 440Hz under damped harmonic motion. The sound is no longer audible once the displacement is less than 0.1mm . What is the damping factor if the sound is no longer audible after 4.0sec ? Round to 2 decimal places.Explain the criteria under which you would choose the model d=sint versus d=acost to represent the displacement d of an object moving in simple harmonic motion.Explain the difference between an object moving in simple harmonic motion versus damped harmonic motion.51PE 52PE 53PE For Exercises 53-54, use the model d=aectcost or d=abktcoswt to represent damped harmonic motion. A pendulum is pulled 18 radians to one side and then released. The angular displacement follows a pattern of damped harmonic motion with each cycle lasting 2sec . If the maximum displacement for each cycle decreases by20 , find a function that models the angular displacement tsec after being released.In a course in differential equations or physics, a more specific model for a mass moving in damped harmonic motion is found as d=aebe/2mcost2b2m2 where b is a damping constant unique to the physical system m is the mass of the object, and |a| is the displacement at time t=0 . The value 2 is the related period under simple harmonic motion. Use this model for Exercises 55-56. a. Write a model for damped harmonic motion for a mass of 5kg oscillating on a spring with initial displacement 0.8m , period of 6sec , and damping constant b=0.3N-sm (Newton-second/meter). b. Graph the model from part (a) on the window 0,10,1by1,1,0.1.In a course in differential equations or physics, a more specific model for a mass moving in damped harmonic motion is found as d=aebe/2mcost2b2m2 where b is a damping constant unique to the 0physical systemm is the mass of the object, and |a| is the displacement at timet=0 . The value 2 is the related period under simple harmonic motion. Use this model for Exercises 55-56. a. Write a model for damped harmonic motion for the bob of a pendulum of mass of 2kg originally pulled to the right of its equilibrium position 0.6m . Upon release, the bob makes one complete swing back and forth in 2sec with a damping constant b=0.2N-sm . b. Graph the model from part (a) on the window 0,10,1by0.8,0.8,0.1.Given the model d2006tcos2t . for an object in damped harmonic motion, a. Determine the initial displacement, d (in centimeters). b. Graph the model on a graphing utility. c. Determine the time t (in seconds) between two consecutive relative maxima. d. By what percentage is the displacement of the object decreased with each successive oscillation between consecutive maxima? Round to the nearest tenth of a percent Given the model d=804tcost for an object in damped harmonic motion, a. Determine the initial displacement, d (in centimeters). b. Graph the model on a graphing utility. c. Determine the time t (in seconds) between two consecutive relative maxima. d. By what percentage is the displacement of the object decreased with each successive oscillation between consecutive maxima? Round to the nearest tenth of a percent.Given a point on the Earth of latitude a, the angle of elevation of the Sun is at its minimum of 9023.5 at noon at the winter solstice. The angle of elevation of the Sun is at its maximum of 90+23.5 at noon on the summer solstice. The angle of elevation follows simple harmonic motion as it varies from its minimum to its maximum over the course of 365 days. Use this information for Exercises 59-60. Dallas, Texas, has latitude 32.8N . a. Find the angle of elevation of the Sun at noon on the winter solstice and summer solstice. b. Suppose that the winter solstice is 10 days before the end of a given year. Write a function representing the angle A (in degrees) of the Sun as a function of the day number t for the following year. (Hint: Use a phase shift of-10 days.) c. Graph the function from part (b) on a graphing utility. d. Determine the angle of elevation of the Sun at noon on February 1 for Dallas. Round to the nearest tenth of a degree. e. How many days into the year will the spring and fall equinoxes occur? Round to the nearest day. (Hint: The angle of elevation of the Sun at noon on an equinox will be halfway between its minimum and maximum value. This is the day of the year when the duration of daylight equals the duration of darkness.)Given a point on the Earth of latitude a, the angle of elevation of the Sun is at its minimum of 9023.5 at noon at the winter solstice. The angle of elevation of the Sun is at its maximum of 90+23.5 at noon on the summer solstice. The angle of elevation follows simple harmonic motion as it varies from its minimum to its maximum over the course of 365 days. Use this information for Exercises 59-60. Denver, Colorado, has latitude39.7N . a. Find the angle of elevation of the Sun at noon on the winter solstice and summer solstice. b. Suppose that the winter solstice is 10 days before the end of a given year. Write a function representing the angle A (in degrees) of the Sun as a function of the day number t for the following year. (Hint Use a phase shift of-10 days.) c. Graph the function from part (b) on a graphing utility. d. Determine the angle of elevation of the Sun at noon on January 15 for Denver. Round to the nearest tenth of a degree.For Exercises 1-2, given a point in polar coordinates, find two other polar coordinate representations, a. with r0 b. with r0 4,23 For Exercises 1-2, given a point in polar coordinates, find two other polar coordinate representations, a. with r0 b. with r0 5,6 For Exercises 3-4, convert the ordered pair in polar coordinates to rectangular coordinates. 6,4 For Exercises 3-4, convert the ordered pair in polar coordinates to rectangular coordinates. 12,2 For Exercises 5-6, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . 9,93 For Exercises 5-6, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . 532,52 For Exercises 7-12, write an equivalent equation using polar coordinates. x+5y=12 For Exercises 7-12, write an equivalent equation using polar coordinates. x=12 For Exercises 7-12, write an equivalent equation using polar coordinates. x2+y32=9 10RE 11RE For Exercises 7-12, write an equivalent equation using polar coordinates. y=x2 For Exercises 13-16, convert the polar equation to rectangular form and identify the type of curve represented. r=16 For Exercises 13-16, convert the polar equation to rectangular form and identify the type of curve represented. r=16cos For Exercises 13-16, convert the polar equation to rectangular form and identify the type of curve represented. =3 For Exercises 13-16, convert the polar equation to rectangular form and identify the type of curve represented. r=10sin 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE For Exercises 27-30, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r . b. The line =2 Replace r, by r, . c. The pole. Replace r, by r, . Otherwise, indicate that the test is inconclusive. r=4cos2 For Exercises 27-30, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r . b. The line =2 Replace r, by r, . c. The pole. Replace r, by r, . Otherwise, indicate that the test is inconclusive. r=12 For Exercises 27-30, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r . b. The line =2 Replace r, by r, . c. The pole. Replace r, by r, . Otherwise, indicate that the test is inconclusive. r=8cos For Exercises 27-30, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r . b. The line =2 Replace r, by r, . c. The pole. Replace r, by r, . Otherwise, indicate that the test is inconclusive. r=8sin For Exercises 31-32, find the modulus of each complex number and write the number in polar form with 02 . 663i 32RE 33RE For Exercises 33-34, convert the complex number from polar form to rectangular form, a+bi . 18cos330+isin330 For Exercises 35-36, given complex numbers z1 and z2 , a. Find z1z2 and write the product in polar form. b. Find z1z2 and write the quotient in polar form. z1=8cos60+isin60,z2=4cos140+isin140 For Exercises 35-36, given complex numbers z1 and z2 , a. Find z1z2 and write the product in polar form. b. Find z1z2 and write the quotient in polar form. z1=10cos512+isin512,z2=15cos4+isin4 For Exercises 37-38, use De Moivreâ€™s theorem to find the indicated power. Write the result in rectangular form a+bi . 2cos12+isin124 For Exercises 37-38, use De Moivreâ€™s theorem to find the indicated power. Write the result in rectangular form a+bi . 3+3i5 For Exercises 39-40, find the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . The four roots of 2401 .For Exercises 39-40, find the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . The six roots of 4096 .For Exercises 41-42, find the indicated complex roots. Write the results in polar form using degree measure for the argument. The square roots of 25cos120+isin120 ,For Exercises 41-42, find the indicated complex roots. Write the results in polar form using degree measure for the argument. The cube roots of 32323i ,43RE 44RE 45RE 46RE For Exercises 47-48, refer to vectors v and w in the figure. Sketch v+w by placing the initial point of w at the terminal point of v.For Exercises 47-48, refer to vectors v and w in the figure. Sketch vw by drawing v and w with the same initial point.For Exercises 49-54, perform the indicated operations for the given vectors. v=3,5w=3,2s=10i6jr=5i+3j 2v+w 50RE 51RE For Exercises 49-54, perform the indicated operations for the given vectors. v=3,5w=3,2s=10i6jr=5i+3j w13v 53RE For Exercises 49-54, perform the indicated operations for the given vectors. v=3,5w=3,2s=10i6jr=5i+3j s+r 55RE 56RE 57RE 58RE 59RE 60RE For Exercises 61-62, the given forces (in units of pounds) act on an object. a. Find the resultant force, R. b. What additional force F is needed for the object to be in static equilibrium? F1=7,3 and F2=10,4 For Exercises 61-62, the given forces (in units of pounds) act on an object. a. Find the resultant force, R. b. What additional force F is needed for the object to be in static equilibrium? F1=8i2j,F2=ij,F3=3i+10j 63RE During a penalty kick, a soccer ball is kicked at an angle of 8 from the horizontal with a speed of53mph . Write the velocity vector v at the instant of the kick in terms of i and j. Round the components to the nearest tenth of a mph.A small plane travels 120mph on a bearing of N40E and encounters a wind of 20 mph acting in the direction of. Rounding to 1 decimal place, a. Express the velocity of the plane v relative to the air in terms of i and j. b. Express the velocity of the wind vw in terms of i and j. c. Find the true velocity vT and true speed of the plane. Round the speed to the nearest mph.66RE For Exercises 66-68, find the indicated value. a vw b. vv v=12i+83j,w=8i34j For Exercises 66-68, find the indicated value. a. vw b. vv v=14i+j,w=80i30j For Exercises 69-72 use the dot product to determine if v and w are orthogonal. If the vectors are not orthogonal, find the angle between them. Round to the nearest tenth of a degree. v=0.2,0.3,w=0.6,0.4 70RE 71RE 72RE Given v=i+3j and w=2ij a. Find projwv . b. Find vectors v1 and v2 such that v1 is parallel to w,v2 is orthogonal to w, and v1+v2=v .Given v=10j and w=9i+12j a. Find projwv . b. Find vectors v1 and v2 such that v1 is parallel to w,v2 is orthogonal to w, and v1+v2=v .A van weighing 2200lb is parked on a street with an 8 incline. a. Write the force vector F representing the weight against a single tire. Write F in terms of i and j and assume that the weight of the van is evenly distributed among all four tires. b. Find the component vector, F1 of F parallel to the street. Round to 1 decimal place. c Find the magnitude of the force required by the brakes on each wheel to keep the truck from rolling down the street. Round to the nearest tenth of a pound.76RE 77RE 78RE 79RE 80RE Convert 5,54 in polar coordinates to rectangular coordinates.Convert 63,6 in rectangular coordinates to polar coordinates withr r0 and 02 .3T For Exercises 3-4, write an equivalent equation using polar coordinates. y=x For Exercises 5-6, write an equivalent equation using rectangular coordinates. r=8 For Exercises 5-6, write an equivalent equation using rectangular coordinates. r=5sec For Exercises 7-10, graph the polar equation and identify the type of curve. r=2cos3 For Exercises 7-10, graph the polar equation and identify the type of curve. r=26cos For Exercises 7-10, graph the polar equation and identify the type of curve. r=5+5sin For Exercises 7-10, graph the polar equation and identify the type of curve. r=5sin2 For Exercises 11-12, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, . b. The line =2 , Replace r, by r, . c. The pole. Replace r, by r, . Otherwise, indicate that the test is inconclusive. r=5cos3 For Exercises 11-12, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, . b. The line =2 , Replace r, by r, . c. The pole. Replace r, by r, . Otherwise, indicate that the test is inconclusive. r=7sin Find the modulus of 55i and write the number in polar form with 02 .Convert 24cos43+isin43 from polar form to rectangular form, a+bi .Find 12cos143+isin1438cos220+isin220 and write the product in polar form.Find 18cos100+isin10016cos55+isin55 and write the quotient in polar form.17T Find the cube roots of 8cos56+isin56 .For Exercises 19-24, perform the indicated operations for v=8,0 and w=5,3 . v+3w For Exercises 19-24, perform the indicated operations for v=8,0 and w=5,3 . v+w For Exercises 19-24, perform the indicated operations for v=8,0 and w=5,3 . 65w For Exercises 19-24, perform the indicated operations for v=8,0 and w=5,3 . 8v10w For Exercises 19-24, perform the indicated operations for v=8,0 and w=5,3 . vw For Exercises 19-24, perform the indicated operations for v=8,0 and w=5,3 . vvw Given v=3,10 , a. Find v2 . b. Find vv . c. How are v2 and vv related?Find a unit vector in the direction of v=8i+6j .27T Find the magnitude and direction angle 0360 for the vector v=2i+7j .Round to 1 decimal place.29T 30T The forces F1=6ij,F2=7i2j and F3=9i+3j act on an object. What additional force F is needed for the object to be in static equilibrium?Given v=7,5 and w=10,12 , Find projwv .A plane heading S21E with an an air speed of 375mph encounters a wind blowing due east at 28mph . a. Express the velocity of the plane vp relative to the air and the velocity of the wind vw in terms of i and j. Round components to 1 decimal place. b. Find the velocity of the plane relative to the ground vg and the speed of the plane relative to the ground. Round the speed to the nearest mph. c. Find the bearing of the plane relative to the ground. Round to the nearest tenth of a degree.34T 35T 36T Find the amount of work done if an object is pushed horizontally 120m by a force of 15N directed 40 above the horizontal. Round to the nearest Nm.For Exercises 1-7, solve for y. Write solutions to inequalities in interval notation. Ax2x+y For Exercises 1-7, solve for y. Write solutions to inequalities in interval notation. z10x2+y2 For Exercises 1-7, solve for y. Write solutions to inequalities in interval notation. sy2+2ty=r For Exercises 1-7, solve for y. Write solutions to inequalities in interval notation. A=A0eky For Exercises 1-7, solve for y. Write solutions to inequalities in interval notation. R10logyy0 For Exercises 1-7, solve for y. Write solutions to inequalities in interval notation. y23y4y+20 For Exercises 1-7, solve for y. Write solutions to inequalities in interval notation. 3y100 8CRE 9CRE Write an equation of a line in slope-intercept form in rectangular coordinates that passes through the points 8,5 and 5,8 .Write an equation of a line in polar coordinates that passes through the points 8,8 and 8,8 .For Exercises 12-14, solve each triangle. Round to the nearest tenth.For Exercises 12-14, solve each triangle. Round to the nearest tenth.For Exercises 12-14, solve each triangle. Round to the nearest tenth.Evaluate the difference quotient fx+hf(x)h for fx=3x .Give the domain and range of fx=3x .Find the period and phase shift of each function. a. y=tan2x+3 b. y=sin2x+3 The height h (in feet) of a swing above ground level at time t (in sec) is modeled by ht=2cos23t+3 . a. Find the maximum and minimum height of the swing. b. When is the first time after t=0 that the swing is at a height of 4 ft? c. What is the period of the swing? d. What is the frequency of the swing?Given v=8,3 and w=12,4 , find vector projection of v onto w.20CRE Plot the points whose polar coordinates are given. a. 3,54 b. 4,76 c. 2.5,4 d. 3,43 Given 7,32 in polar coordinates, find two other polar coordinate representations. a. with r0 b. with r0 Convert the ordered pairs in polar coordinates to rectangular coordinates. a. 5, b. 10,23 Convert 2,23 from rectangular coordinates to polar coordinates. Give two representations, one with r0 and one with r0 .5SP Write an equivalent equation using polar coordinates. a. y=3 b. y=x2 1PE Given a point P represented by the ordered pair r, in polar coordinates, the distance from P to 0,0 is .To convert an ordered pair r, in polar coordinates to rectangular coordinates, use the relationships x= and y= .To convert an ordered pair x,y in polar coordinates to rectangular coordinates, use the relationships r2= and tan= .Use an equation in polar coordinates to describe the set of points 4 units from the pole.6PE 7PE For Exercises 7-14, match each point in polar coordinates to the points A, B, C. or D. A 3,6 B 3,34 C 3,6 D 3,4 3,116 For Exercises 7-14, match each point in polar coordinates to the points A, B, C. or D. A 3,6 B 3,34 C 3,6 D 3,4 3,116 For Exercises 7-14, match each point in polar coordinates to the points A, B, C. or D. A 3,6 B 3,34 C 3,6 D 3,4 3,54 For Exercises 7-14, match each point in polar coordinates to the points A, B, C. or D. A 3,6 B 3,34 C 3,6 D 3,4 3,134 For Exercises 7-14, match each point in polar coordinates to the points A, B, C. or D. A 3,6 B 3,34 C 3,6 D 3,4 3,74 For Exercises 7-14, match each point in polar coordinates to the points A, B, C. or D. A 3,6 B 3,34 C 3,6 D 3,4 3,56 For Exercises 7-14, match each point in polar coordinates to the points A, B, C. or D. A 3,6 B 3,34 C 3,6 D 3,4 3,76 15PE For Exercises 15-18, plot the points whose polar coordinates are given. (See Example 1) A 3,3 B 4.5,114 C 5,56 D 3,4 17PE For Exercises 15-18, plot the points whose polar coordinates are given. (See Example 1) a. 2,23 b. 3, c. 4,4 d. 5,56 For Exercises 19-22, given a point in polar coordinates, find two other polar coordinate representations. (See Example 2) a. with r0 . b. with r0 . 7,56 For Exercises 19-22. given a point in polar coordinates, find two other polar coordinate representations. (See Example 2) a with r0 .b with r0 . 3.6,2 For Exercises 19-22, given a point in polar coordinates, find two other polar coordinate representations. (See Example 2) a. with r0 . b. with r0 . 8,74 For Exercises 19-22, given a point in polar coordinates, find two other polar coordinate representations. (See Example 2) a. with r0 . b. with r0 . 2,43 For Exercises 23-32, convert the ordered pair in polar coordinates to rectangular coordinates. (See Example 3) 8,56 For Exercises 23-32, convert the ordered pair in polar coordinates to rectangular coordinates. (See Example 3) 6,23 25PE For Exercises 23-32, convert the ordered pair in polar coordinates to rectangular coordinates. (See Example 3) 34,74 For Exercises 23-32, convert the ordered pair in polar coordinates to rectangular coordinates. (See Example 3) 17,3 For Exercises 23-32, convert the ordered pair in polar coordinates to rectangular coordinates. (See Example 3) 5,76 For Exercises 23-32, convert the ordered pair in polar coordinates to rectangular coordinates. (See Example 3) 4,116 For Exercises 23-32, convert the ordered pair in polar coordinates to rectangular coordinates. (See Example 3) 8,43 For Exercises 23-32, convert the ordered pair in polar coordinates to rectangular coordinates. (See Example 3) 7,2 32PE 33PE For Exercises 33-38, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . (See Example 4-5) 73,7 For Exercises 33-38, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . (See Example 4-5) 82,82 36PE 37PE For Exercises 33-38, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . (See Example 4-5) 4,10 For Exercises 39-44, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . (See Examples 4-5) 532,52 For Exercises 39-44, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . (See Examples 4-5) 322,322 41PE For Exercises 39-44, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . (See Examples 4-5) 102,102 For Exercises 39-44, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . (See Examples 4-5) 3,8 For Exercises 39-44, convert the ordered pair in rectangular coordinates to polar coordinates with r0 and 02 . (See Examples 4-5) 5,1 45PE 46PE For Exercises 45-56, write an equivalent equation using polar coordinates. (See Example 6) x=6 For Exercises 45-56, write an equivalent equation using polar coordinates. (See Example 6) y=2 49PE 50PE For Exercises 45-56, write an equivalent equation using polar coordinates. (See Example 6) x2=3y 52PE 53PE 54PE 55PE 56PE For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=3 For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=7 For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=2csc For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=sec For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=4cos For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=2sin 63PE For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) =6 For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=asina0 66PE 67PE For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=acosa0 For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=2hcos+2ksin For Exercises 57-70, convert the polar equation to rectangular form and identify the type of curve represented. (See Example 7) r=hcos+ksin Suppose that the polar axis is positioned to coincide with the positive x-axis in a rectangular coordinate system. Describe the graphs of the polar equations as they relate to a rectangular coordinate system. a. =0 b. =4 c. =2 72PE

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