Arc Length Lesson Plan

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1 Ac Length Leon Plan By: Dougla A. Ruby Date: 10/10/2002 Cla: Pe-Calculu II Gade: 11/12 INSTRUCTIONAL OBJECTIVES: At the end of thi leon, the tudent will be able to: 1. Find the length of an ac of a cicle, given the meaue of it cental angle and the length of a adiu. 2. Find the meaue of a cental angle of a cicle, given the length of it ac and the length of a adiu. 3. Convet between linea and angula peed 4. Find total ditance o total angle, when peed, adiu, and time ae given. Relevant Maachuett Cuiculum Famewok PC.M.1 Decibe the elationhip between degee and adian meaue, and ue adian meaue in the olution of poblem, paticulaly poblem involving angula velocity and acceleation. PC.P.3 - Demontate an undetanding of the tigonometic function (ine, coine, tangent, coecant, ecant, and cotangent). Relate the function to thei geometic definition. MENTAL MATH (5 Minute) Quetion 1: Find the eal numbe M, N, P, and Q between -2 and 2 point on the unit cicle. that detemine each of the M=2 /3 N=5 /6 P=5 /4 P=11 /6 Quetion 2: What i the value of the ine function fo the point (.5, ) on the unit cicle? What i the angle in decibed by thi point? 60 o o /3 adian Page 1

2 Ac Length Leon Plan M. Ruby CLASS ACTIVITIES (Note: 45 Minute Leon Plan) 1. Ac Length aound a cental Angle. In the peviou leon, we coveed the notion of adian elative to the unit cicle. Radian meaue can be detemined uing a cicle othe than a unit cicle. In the following figue, a unit cicle i hown along with anothe cicle. The angle hown i a cental angle of both cicle. By vitue of the fact that the elationhip between the cicumfeence c of a cicle and it diamete d i linea: c = d It hould be appaent that the adii of the two cicle ( and 1) and the ac ( 1 and ) ub-tended by the cental angle ae in diect popotion. Thi mean that we can wite a popotion that define the elationhip between the ac of the unit cicle and the ac of any cicle, auming that we have a common cental angle. Thi elationhip i: 1 1 o by co multiplying, 1 1 Now conide fom ou pio dicuion on the unit cicle, that the ac 1 i actually jut the adian meaue of the cental angle! Thu, we can call 1 by one of the Geek ymbol fo angle, uch a. Theefoe, when we want to know the angle ubtended by any ac of any cicle of adiu, we can jut ue the equation: whee i meaued in adian! Of coue, thi make adian meaue vey eay to ue when dealing with ditance aound ac of cicle. Keep in mind, that when we ue the fomula above, i alway meaued in adian, and and mut have the ame unit of meaue. Let ty an example: Example: Find the length of an ac of a cicle with 10-cm adiu aociated with a cental angle meaue of 11 /6 adian. Solution: In ou example, we know that the angle = 11 /6 adian and that the adiu = 10cm.Thu, we need to olve fo the ac. In the above equation, =, o: = 11 /6 10cm. = 55 /3 cm. Page 2

3 Ac Length Leon Plan M. Ruby Let ty ome example on you own 1. Find the adian meaue of a otation whee a point 2.5 cm fom the cente of otation tavel 15 cm. = 15cm./2.5cm. =6 adian 2. Find the adian meaue of a otation whee a point 24 in. fom the cente of otation tavel 3 ft. 2. Angula Speed = 36in./24in. = 1.5 adian Nomally, we think of peed: in tem of unit like mile pe hou, o ome fom of ditance pe unit time. Similaly, angula peed i defined to be amount of otation pe unit of time. Fo example, we might peak of the angula peed of a wheel a 150 evolution pe minute, o the angula peed of the eath a 2 adian pe day. The ymbol (omega) i geneally ued fo angula peed. Thu angula peed i defined a = t Let take thi a bit futhe. Suppoe you wanted to know the linea peed of the Space Station in otation aound the eath at a cetain ditance fom the axi of otation (adiu )? Well, to deive the fomula fo linea peed let fit look at the ditance of an ac. Remembe, that the length (ditance) of an ac i: = So to know the peed o velocity v, we need to know the ditance pe unit time. Thi would be: v t t but ince =, we can ubtitute and get: v = t Thu, the linea peed of a body in otation i equal to the angula peed (in adian/unit time) time the adiu of otation. Let ty an example: Example: A atellite in eath obit 1200km above the eath make a otation evey 90 min. Given that the adiu of the eath i 6400km., what i the linea peed of the atellite in km/h? Solution: Ok, what ae ou data point? 1. The total adiu = 1200km km. = 7600km. 2. Angula peed = 2 adian pe 90 min. = h. = 4 /3 adian/h Page 3

4 Ac Length Leon Plan M. Ruby 3. Linea peed v = 4 /3 adian/h 7600km = 31,835 km/h. Cool, eh? Let ty ome poblem on you own: 1. A wheel with a 12-cm. diamete i otating at a peed of 10 evolution pe econd. What i the velocity v of a point on the im in cm/ec? = 6 cm., = 20 /ec., v = 120 cm./ec. 2. An old oaken bucket i being aied fom a well at a ate of 3 ft/ec. The adiu of the dum the ope i being wapped aound i 5 in. What i the angula peed of the dum in evolution pe econd? = 5 in., v = 3ft./ec=.36in/ec., =.36/5= 7.2 adian/ec.= 7.2/ ev/ec. 3. Total Ditance and Total Angle Finally, let look at poblem involving total ditance and total angle. Both velocity v (aka. linea peed) and angula peed ae known a ate. A ate i a meauement pe unit of time. Thu, in ode to know the total ditance d a point tavel, o the total angle that the point otate, we need to know the total time t. By knowing t we get: d = v t and = t Let ty an example: Example: The eath ha a 4000-mi adiu and otate one evolution evey 24 h. What i the linea peed of a point on the equato, in mile pe hou? How fa doe that point on the equato tavel in 4 day?. = 1 ev/day 1/24 day/h. 2 ad./ev. =.2618 ad./h. v= =.2618 ad.h mi. = 1047 mph. d = 1047 mph. 4 day 24 h./day = mile 4. Home wok dicuion (5 Minute) Page 4

5 Ac Length Leon Plan M. Ruby HOMEWORK (Mateial): Souce: - Bittinge and Beeche, Algeba and Tigonomety, Section 6.4, pp Joyce, David E., Dave' Shot Tig Coue, Section 2, 9, and 10 Solve the attached homewok poblem. Requie TI-83+ o equivalent Algebaic (gaphing) calculato 1. In a cicle with 120-cm adiu, an ac 132 cm long ub-tend a cental angle of how many adian? How many degee, to the neaet degee? = / = 132 cm/120 cm = 1.1 adian, 1.1 adian 180/ deg/adian 63 o 2. In a cicle with 200-cm adiu, an ac 65 cm long ub- tend a cental angle of how many adian? How many degee, to the neaet degee? = / =65 cm/200 cm =.325 adian,.325 adian 180/ deg/adian 19 o 3. Though how many adian doe the minute hand of a clock otate in 50 min? =50/60 ev. 2 ad./ev. = 5.24 adian 4. A wheel on a ca ha a 14-in. adiu. Though what angle (in adian) doe the wheel tun while the ca tavel 1 mi? =((5280 ft. 12 in./ft.) (14in. 2 )) evolution 2 ad/ev. = 4526 adian 5. In a cicle with 10-m adiu, how long i an ac aociated with a cental angle of 1.6 adian? = /, = = 10m 1.6 ad. = 16m. 6. In a cicle with 5-m adiu, how long i an ac aociated with a cental angle of 2.1 adian? = /, = = 5m 2.1 ad. = 10.5m. 7. A flywheel with 15-cm diamete i otating at a ate of 7 adian/ec. What i the linea peed of a point on it im, in centimete pe minute? Note: 15-cm diamete = 7.5cm adiu 7 ad./ec = 7ad./ec 60 ec./min. = 420 ad./min, 420 ad./min 7.5cm. = 3150 cm./min. 8. A wheel with 30-cm adiu i otating at a ate of 3 adian/ec. What i the linea peed of a point on it im, in mete pe minute? 3 ad./ec = 3 ad./ec 60 ec./min. = 180 ad./min, 180 ad./min 30 cm. = 5400 cm./min. = 54 m./min 9. A 33-pm ecod ha a adiu of 15 cm. What i the linea velocity of a point on the im, in centimete pe econd? Note: 33 1/3pm = 100/3 pm = 100/3 pm 2 ad./ev. 1/60 min./ec. = 3.49 ad./ec. v= = 3.49 ad./ec. 15 cm. = cm./ec. 10. A 45-pm ecod ha a adiu of 8.7 cm. What i the linea velocity of a point on the im, in centimete pe econd? = 45 pm 2 ad./ev. 1/60 min./ec. =3 /2 ad./ec. v= = 3 /2 ad./ec. 8.7 cm. = 41 cm./ec. Page 5

6 Ac Length Leon Plan M. Ruby 11. The eath i 93,000,000 mile fom the un and tavee it obit, which i nealy cicula, evey day. What i the linea velocity of the eath in it obit, in mile pe hou? v= = 93,000,000 mi. (1 ev/y. 2 ad./ev) ( day/y. 24 h./day) = 66,659 mph. 12. A wheel ha a 32-cm diamete. The peed of a point on it im i 11 m/. What i it angula peed? = v / = 11 m./. 100 cm./m. / 32 cm. = ad./ec. 13. A hoe on a mey-go-ound i 7 m fom the cente and tavel at a peed of 10 km/h. What i it angula peed? = v / = 10 km./h m./km. / 7 m. = ad./h. Bonu Poblem (exta cedit) 14. Find the length to the neaet inch of a cicula ac of 11 degee 48.3 minute if the adiu i 3200 feet Solution: 11 * *3200 ft ft " A aiload cuve fom a cicula ac of 9 degee 36.7 minute, the adiu to the centeline of the tack being 2100 feet. If the gauge i 5 feet, find the diffeence in length of the two ail to the neaet half-inch. Solution: * * 2100 ft 2.5 ft 9 * * (2100 ft 2.5 ft) ft Page 6

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