Lesson 8.1 Graphing Parametric Equations


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1 Lesson 8.1 Grphing Prmetric Equtions 1. rete tle for ech pir of prmetric equtions with the given vlues of t.. x t 5. x t 3 c. x t 1 y t 1 y t 3 y t t t {, 1, 0, 1, } t {4,, 0,, 4} t {4, 0,, 4, 8}. Find four points on the grph of ech pir of prmetric equtions. (hoose your own vlues of t.). x 3t 1, y t 1. x t, y t 5 c. x t, y t d. x t, y t t e. x t, y t f. x t 3 4, y (t ) 3. Without ctully grphing, determine whether the grph of ech pir of prmetric equtions is line, prol, or semicircle. (ssume tht the tintervl llows the complete grph to e trced.). x t, y t 3t 1. x t, y 4 t c. x t, y t 1 d. x t 1, y t 1 e. x t 3, y t 1 f. x t, y 1 t 4. Roz nd Din re oth tking wlks. These prmetric equtions simulte their motion, with x nd y mesured in meters nd t mesured in seconds. Roz Din x 1.t x 380 y 50 y 1.1t. Grph the motion of the wlkers for 0 t 500 in window tht shows ll of their wlk.. Give relworld mening for ech vlue in the equtions. c. Where do the two pths meet? d. Do Roz nd Din meet? How do you know? Discovering dvnced lger More Prctice Your Skills HPTER 8 49
2 Lesson 8. onverting from Prmetric to Nonprmetric Equtions 1. Solve ech eqution for t.. x t 1. y t 3 c. x t 1 d. y 9 t e. y t t 1 f. x 5 t 3. Write single eqution using only x nd y tht is equivlent to ech pir of prmetric equtions. (Ech eqution should express y in terms of x.). x t. x t 5 c. x t y 3t t 1 y t 3 y 3t d. x t 7 e. x t 4 f. x t 3 y t 5 y 3(t ) y t 5 g. x t 1 h. x t 1 i. 4 x t 4 y t 3 y 3 t 1 y 3t 5 3. This tle gives x nd yvlues for severl vlues of t.. Write n eqution for x in terms of t.. Write n eqution for y in terms of t. c. Eliminte the prmeter nd comine the two equtions. 4. Use the grphs of x f(t) nd y g(t) to crete grph of y s function of x. x y t x y t t 50 HPTER 8 Discovering dvnced lger More Prctice Your Skills
3 Lesson 8.3 Right Tringle Trigonometry 1. For ech of the following right tringles, find the vlues of sin, cos, tn nd sin, cos, tn. (Write your nswers s frctions in lowest terms.) Drw right tringle for ech prolem. Lel the sides nd ngles with the given mesures, then solve to find the unknown vlue. Round your nswers to the nerest tenth. r. cos tn S c. sin 81 w d. 1.8 cos 5 z 3 z 3. For ech tringle, write n eqution to clculte the leled mesure. Then solve the eqution....9 c Drw pir of horizontl nd verticl lines, nd lel the four directions with N, E, S, nd W for north, est, south, nd west. Sketch the pth of plne flying on ering of 74.. Find the mesure of the ngle etween the plne s pth nd the horizontl xis.. Write pir of prmetric equtions tht descrie the plne s position fter flying for t hours long this pth t 375 mi/h. c. If the plne flies t 375 mi/h for.5 h, how fr est nd how fr north will it hve trveled? Discovering dvnced lger More Prctice Your Skills HPTER 8 51
4 Lesson 8.4 Using Trigonometry to Set ourse 1. Find the ering of ech direction. Give the signs of the component vectors for ech ering.. 37 est of north. 37 north of est c. 54 west of north d. 54 north of west e. 4 west of south f. 4 south of west g. 11 south of est h. 11 est of south i. 11 north of west. Drw compss rose nd vector with mgnitude v for ech ering. Find the ngle the vector mkes with the xxis c. 3 d Imgine tht cr is moving with the given speed t the given ngle. Write prmetric equtions for the horizontl nd verticl components of ech motion in terms of the sine nd cosine of n cute ngle.. 15 units/s t 4 from the xxis. 3 units/s t 7 from the yxis c. 1 units/s on ering of 1 d. 1 units/s on ering of 303 e. 39 units/s on ering of Give the ering for the ctul course flown y ech irplne nd the ctul distnce tht the irplne will trvel in 3 hours. (Round ngles to the nerest tenth of degree nd distnces to the nerest mile.). The pilot heds the plne due est t 300 mi/h. There is constnt 5 mi/h wind lowing from the south.. The pilot heds the plne due west t 50 mi/h. There is constnt 30 mi/h wind lowing from the south. c. The pilot heds the plne due west t 38 mi/h. There is constnt mi/h wind lowing from the north. d. The pilot heds the plne due est t 404 mi/h. There is constnt 3.5 mi/h wind lowing from the north. 5 HPTER 8 Discovering dvnced lger More Prctice Your Skills
5 Lesson 8.5 Projectile Motion 1. The height of flling oject is given y the eqution y 1t s 0, where t is the time (mesured in seconds) nd s 0 is the initil height of the oject (mesured in feet). Find the height of n oject dropped from the indicted height fter the indicted mount of time.. s 0 5, t 1.5. s 0 100, t.4 c. s 0 4, t 4.7. Find ll positive solutions for ech eqution. Round to the nerest hundredth.. 1t t c. 4.9t 9.8t 30 0 d. 1t 3t 55 0 e. 1t 100t sin f. 4.9t 35t sin Write prmetric equtions to simulte ech motion.. ll rolls off the edge of tle tht is 1.75 m high t n initil velocity of.3 m/s.. sell is hit with n initil velocity of 13 ft/s t n ngle of 8. The t contcts the ll t height of 3.5 ft ove the ground. c. golfer swings clu with loft of 4 nd n initil velocity of 18 ft/s on level ground. 4. In ech sitution descried in Exercise 3, how long will it tke the ll to hit the ground? Round to the nerest hundredth. 5. Hyden rolls ll off the edge of the roof of 75 ft tll uilding t n initil velocity of.5 ft/s.. Write prmetric equtions to simulte this motion.. Wht eqution cn you solve to determine when the ll hits the ground? c. How long fter it rolls off the roof does the ll hit the ground? (Round to the nerest hundredth.) d. How fr from the se of the uilding (directly elow the spot where the ll rolls off the roof) does the ll hit the ground? (Round to the nerest hundredth.) Discovering dvnced lger More Prctice Your Skills HPTER 8 53
6 Lesson 8. The Lw of Sines 1. Solve ech eqution for. Give n exct nswer nd n pproximte nswer. Round to the nerest tenth.. sin 75 9 sin 0. sin 95 sin 4. 5 c. sin sin Find the unknown ngle mesures nd side lengths. Round to the nerest tenth cm mm 3. Determine the numer of tringles with the given prts. (Do not find the missing side lengths nd ngle mesures.). 3, 48, 13.5 cm. 103, 8.5 cm,.7 cm c. 50,. cm, c 7. cm d. 8, 5 cm, c 3 cm 4. ship is siling due est. t certin point, the cptin oserves lighthouse t ering of 3. fter the ship sils 8.0 km frther, the ering of the sme lighthouse from the ship is Drw digrm to illustrte this sitution. Let e the point where the first lighthouse oservtion is mde, e the point where the second oservtion is mde, nd e the loction of the lighthouse. Show the given erings nd distnce on your digrm.. Find the mesure of ech ngle in. 1 c. Use the Lw of Sines to write equtions for the distnce etween the lighthouse nd the ship t the time of the first oservtion nd t the time of the second oservtion. d. Find the distnce etween the lighthouse nd the ship t the time of the first oservtion. e. Find the distnce etween the lighthouse nd the ship t the time of the second oservtion. c 54 HPTER 8 Discovering dvnced lger More Prctice Your Skills
7 Lesson 8.7 The Lw of osines 1. Solve for nd. ssume is positive (14)(9) cos (4.0)(5.1) cos. Use the given vlues to write n eqution for the unknown mesure. Then solve the eqution. Give n exct nswer nd n pproximte nswer. Round to the nerest tenth c. 3.5 c c c c 3. Stte whether you would use the Lw of Sines or the Lw of osines to solve ech prolem.. Given the mesures of two ngles of tringle nd the length of one of the sides tht is not etween them, find the length of one of the other two sides.. Given the lengths of ll three sides of tringle, find the mesure of the smllest ngle. c. Given the lengths of two sides nd the mesure of the ngle etween them, find the length of the third side. d. Given the mesures of two ngles of tringle nd the length of the side etween them, find the length of one of the other two sides. 4. Find unknown ngle mesures nd side lengths. Round to the nerest tenth cm c 1.8 cm.9 cm 7.5 cm 3.8 cm 5. Use the Lw of osines to solve ech prolem. Round to the nerest tenth.. Wht is the length of the longer digonl of prllelogrm with sides of length 13. cm nd 18.9 cm nd with pir of opposite ngles tht ech mesure 47?. Wht is the mesure of the lrgest ngle of tringulr grden whose sides mesure 1.5 ft, 19.8 ft, nd 15.7 ft? c. Sr nd Geoff hiked.5 mi on ering of 58. Then they turned nd hiked n dditionl 3.1 mi on ering of 158. How fr were they from their strting point? Discovering dvnced lger More Prctice Your Skills HPTER 8 55
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