4.8 EXERCISES. VOCABULARY: Fill in the blanks. SKILLS AND APPLICATIONS

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1 Section 4.8 pplictions nd Models EXERCISES See for worked-out solutions to odd-numered exercises. VOCULRY: Fill in the lnks. 1. mesures the cute ngle pth or line of sight mkes with fixed north-south line.. point tht moves on coordinte line is sid to e in simple if its distnce d from the origin t time t is given y either d sin t or d cos t. 3. The time for one complete cycle of point in simple hrmonic motion is its. 4. The numer of cycles per second of point in simple hrmonic motion is its. SKILLS ND PPLICTIONS In Exercises 5 14, solve the right tringle shown in the figure for ll unknown sides nd ngles. Round your nswers to two deciml plces , , 7. 71, , 9. 3, 4. 5, , c , , , C In Exercises 15 18, find the ltitude of the isosceles tringle shown in the figure. Round your nswers to two deciml plces. 45, , c c , 7, LENGTH The sun is 5 ove the horizon. Find the length of shdow cst y uilding tht is 0 feet tll (see figure). c c 35 c LENGTH The sun is 0 ove the horizon. Find the length of shdow cst y prk sttue tht is 1 feet tll. 1. HEIGHT ldder 0 feet long lens ginst the side of house. Find the height from the top of the ldder to the ground if the ngle of elevtion of the ldder is 80.. HEIGHT The length of shdow of tree is 15 feet when the ngle of elevtion of the sun is 33. pproximte the height of the tree. 3. HEIGHT From point 50 feet in front of church, the ngles of elevtion to the se of the steeple nd the top of the steeple re 35 nd 47 40, respectively. Find the height of the steeple. 4. DISTNCE n oserver in lighthouse 350 feet ove se level oserves two ships directly offshore. The ngles of depression to the ships re 4 nd 6.5 (see figure). How fr prt re the ships? 350 ft Not drwn to scle 5. DISTNCE pssenger in n irplne t n ltitude of kilometers sees two towns directly to the est of the plne. The ngles of depression to the towns re 8 nd 55 (see figure). How fr prt re the towns? ft km 5 Not drwn to scle

2 358 Chpter 4 Trigonometry 6. LTITUDE You oserve plne pproching overhed nd ssume tht its speed is 550 miles per hour. The ngle of elevtion of the plne is 16 t one time nd 57 one minute lter. pproximte the ltitude of the plne. 7. NGLE OF ELEVTION n engineer erects 75-foot cellulr telephone tower. Find the ngle of elevtion to the top of the tower t point on level ground 50 feet from its se. 8. NGLE OF ELEVTION The height of n outdoor sketll ckord is 1 1 feet, nd the ckord csts shdow feet long. () Drw right tringle tht gives visul representtion of the prolem. Lel the known nd unknown quntities. () Use trigonometric function to write n eqution involving the unknown quntity. (c) Find the ngle of elevtion of the sun. 9. NGLE OF DEPRESSION cellulr telephone tower tht is 150 feet tll is plced on top of mountin tht is 100 feet ove se level. Wht is the ngle of depression from the top of the tower to cell phone user who is 5 horizontl miles wy nd 400 feet ove se level? 30. NGLE OF DEPRESSION Glol Positioning System stellite orits 1,500 miles ove Erth s surfce (see figure). Find the ngle of depression from the stellite to the horizon. ssume the rdius of Erth is 4000 miles mi Not drwn to scle ,500 mi ngle of depression GPS stellite () Find the length l of the tether you re holding in terms of h, the height of the lloon from top to ottom. () Find n expression for the ngle of elevtion from you to the top of the lloon. (c) Find the height h of the lloon if the ngle of elevtion to the top of the lloon is HEIGHT The designers of wter prk re creting new slide nd hve sketched some preliminry drwings. The length of the ldder is 30 feet, nd its ngle of elevtion is 60 (see figure). 30 ft h 60 () Find the height h of the slide. () Find the ngle of depression from the top of the slide to the end of the slide t the ground in terms of the horizontl distnce d the rider trvels. (c) The ngle of depression of the ride is ounded y sfety restrictions to e no less thn 5 nd not more thn 30. Find n intervl for how fr the rider trvels horizontlly. 33. SPEED ENFORCEMENT police deprtment hs set up speed enforcement zone on stright length of highwy. ptrol cr is prked prllel to the zone, 00 feet from one end nd 150 feet from the other end (see figure). d Enforcement zone 31. HEIGHT You re holding one of the tethers ttched to the top of gint chrcter lloon in prde. efore the strt of the prde the lloon is upright nd the ottom is floting pproximtely 0 feet ove ground level. You re stnding pproximtely 0 feet hed of the lloon (see figure). 00 ft Not drwn to scle l 150 ft l 3 ft 0 ft 0 ft h Not drwn to scle () Find the length l of the zone nd the mesures of the ngles nd (in degrees). () Find the minimum mount of time (in seconds) it tkes for vehicle to pss through the zone without exceeding the posted speed limit of 35 miles per hour.

3 Section IRPLNE SCENT During tkeoff, n irplne s ngle of scent is 18# nd its speed is 75 feet per second. N W () Find the plne s ltitude fter 1 minute. () How long will it tke the plne to clim to n ltitude of,000 feet? 35. NVIGTION n irplne flying t 600 miles per hour hs ering of 5#. fter flying for 1.5 hours, how fr north nd how fr est will the plne hve trveled from its point of deprture? 36. NVIGTION jet leves Reno, Nevd nd is heded towrd Mimi, Florid t ering of 0#. The distnce etween the two cities is pproximtely 47 miles. () How fr north nd how fr west is Reno reltive to Mimi? () If the jet is to return directly to Reno from Mimi, t wht ering should it trvel? 37. NVIGTION ship leves port t noon nd hs ering of S 9# W. The ship sils t 0 knots. () How mny nuticl miles south nd how mny nuticl miles west will the ship hve trveled y 6:00 P.M.? () t 6:00 P.M., the ship chnges course to due west. Find the ship s ering nd distnce from the port of deprture t 7:00 P.M. 38. NVIGTION privtely owned ycht leves dock in Myrtle ech, South Crolin nd heds towrd Freeport in the hms t ering of S 1.4# E. The ycht verges speed of 0 knots over the 48 nuticl-mile trip. () How long will it tke the ycht to mke the trip? () How fr est nd south is the ycht fter 1 hours? (c) If plne leves Myrtle ech to fly to Freeport, wht ering should e tken? 39. NVIGTION ship is 45 miles est nd 30 miles south of port. The cptin wnts to sil directly to port. Wht ering should e tken? 40. NVIGTION n irplne is 160 miles north nd 85 miles est of n irport. The pilot wnts to fly directly to the irport. Wht ering should e tken? 41. SURVEYING surveyor wnts to find the distnce cross swmp (see figure). The ering from to is N 3# W. The surveyor wlks 50 meters from, nd t the point C the ering to is N 68# W. Find () the ering from to C nd () the distnce from to. 359 pplictions nd Models E S C 50 m LOCTION OF FIRE Two fire towers re 30 kilometers prt, where tower is due west of tower. fire is spotted from the towers, nd the erings from nd re N 76# E nd N 56# W, respectively (see figure). Find the distnce d of the fire from the line segment. N W E S 56 d km Not drwn to scle GEOMETRY In Exercises 43 nd 44, find the ngle # etween two nonverticl lines L1 nd L. The ngle # stisfies the eqution tn # $! m " m1 1 1 m m1! where m1 nd m re the slopes of L1 nd L, respectively. (ssume tht m1m! "1.) 43. L1: 3x & y! 5 L: x ' y! L1: x & y! 8 L: x & 5y! &4 45. GEOMETRY Determine the ngle etween the digonl of cue nd the digonl of its se, s shown in the figure GEOMETRY Determine the ngle etween the digonl of cue nd its edge, s shown in the figure.

4 360 Chpter 4 Trigonometry 47. GEOMETRY Find the length of the sides of regulr pentgon inscried in circle of rdius 5 inches. 48. GEOMETRY Find the length of the sides of regulr hexgon inscried in circle of rdius 5 inches. 49. HRDWRE Write the distnce y cross the flt sides of hexgonl nut s function of r (see figure) OLT HOLES The figure shows circulr piece of sheet metl tht hs dimeter of 40 centimeters nd contins 1 eqully-spced olt holes. Determine the stright-line distnce etween the centers of consecutive olt holes. TRUSSES In Exercises 51 nd 5, find the lengths of ll the unknown memers of the truss r 60 x c y 36 ft 40 cm HRMONIC MOTION In Exercises 53 56, find model for simple hrmonic motion stisfying the specified conditions. Displcement mplitude Period t centimeters seconds meters 6 seconds inches 3 inches 1.5 seconds 56. feet feet seconds HRMONIC MOTION In Exercises 57 60, for the simple hrmonic motion descried y the trigonometric function, find () the mximum displcement, () the frequency, (c) the vlue of d when t 5, nd (d) the lest positive vlue of t for which d 0. Use grphing utility to verify your results. 9 ft 30 6 ft 6 ft 35 cm 57. d 9 cos t 59. d 1 sin 6t TUNING FORK point on the end of tuning fork moves in simple hrmonic motion descried y d sin t. Find given tht the tuning fork for middle C hs frequency of 64 virtions per second. 6. WVE MOTION uoy oscilltes in simple hrmonic motion s wves go pst. It is noted tht the uoy moves totl of 3.5 feet from its low point to its high point (see figure), nd tht it returns to its high point every seconds. Write n eqution tht descries the motion of the uoy if its high point is t t 0. Equilirium 63. OSCILLTION OF SPRING ll tht is oing up nd down on the end of spring hs mximum displcement of 3 inches. Its motion (in idel conditions) is modeled y y 1 4 cos 16t t > 0, where y is mesured in feet nd t is the time in seconds. () Grph the function. () Wht is the period of the oscilltions? (c) Determine the first time the weight psses the point of equilirium y NUMERICL ND GRPHICL NLYSIS The cross section of n irrigtion cnl is n isosceles trpezoid of which 3 of the sides re 8 feet long (see figure). The ojective is to find the ngle tht mximizes the re of the cross section. Hint: The re of trpezoid is h 1. Low point 8 ft 8 ft 8 ft d 1 cos 0t d 1 sin 79t 64 High point 3.5 ft

5 Section 4.8 pplictions nd Models 361 () Complete seven dditionl rows of the tle. se 1 se ltitude re cos 8 sin cos 0 8 sin () Use grphing utility to generte dditionl rows of the tle. Use the tle to estimte the mximum cross-sectionl re. (c) Write the re s function of. (d) Use grphing utility to grph the function. Use the grph to estimte the mximum cross-sectionl re. How does your estimte compre with tht of prt ()? 65. NUMERICL ND GRPHICL NLYSIS -meter-high fence is 3 meters from the side of grin storge in. grin elevtor must rech from ground level outside the fence to the storge in (see figure). The ojective is to determine the shortest elevtor tht meets the constrints. () Complete four rows of the tle m m () Use grphing utility to generte dditionl rows of the tle. Use the tle to estimte the minimum length of the elevtor. (c) Write the length L 1 L s function of. (d) Use grphing utility to grph the function. Use the grph to estimte the minimum length. How does your estimte compre with tht of prt ()? 66. DT NLYSIS The tle shows the verge sles S (in millions of dollrs) of n outerwer mnufcturer for ech month t, where t 1 represents Jnury. L L 1 L 1 L L 1 L sin 0.1 sin 0. 3 cos cos Time, t Sles, S Time, t Sles, S () Crete sctter plot of the dt. () Find trigonometric model tht fits the dt. Grph the model with your sctter plot. How well does the model fit the dt? (c) Wht is the period of the model? Do you think it is resonle given the context? Explin your resoning. (d) Interpret the mening of the model s mplitude in the context of the prolem. 67. DT NLYSIS The numer of hours H of dylight in Denver, Colordo on the 15th of ech month re: 19.67,.7, 311.9, 413.5, , , 714.7, , 91.48, 11.18, 11.00, The month is represented y t, with t 1 corresponding to Jnury. model for the dt is given y Ht sint () Use grphing utility to grph the dt points nd the model in the sme viewing window. () Wht is the period of the model? Is it wht you expected? Explin. (c) Wht is the mplitude of the model? Wht does it represent in the context of the prolem? Explin. EXPLORTION 68. CPSTONE While wlking cross flt lnd, you notice wind turine tower of height h feet directly in front of you. The ngle of elevtion to the top of the tower is degrees. fter you wlk d feet closer to the tower, the ngle of elevtion increses to degrees. () Drw digrm to represent the sitution. () Write n expression for the height h of the tower in terms of the ngles nd nd the distnce d. TRUE OR FLSE? In Exercises 69 nd 70, determine whether the sttement is true or flse. Justify your nswer. 69. The Lening Tower of Pis is not verticl, ut if you know the ngle of elevtion to the top of the tower when you stnd d feet wy from it, you cn find its height h using the formul h d tn. 70. N 4 E mens 4 degrees north of est.