Section 2.2 Arc Length and Sector Area. Arc Length. Definition. Note:

Transcription

1 Section. Arc Length and Sector Area Arc Length Definition If a central angle, in a circle of a radiu r, cut off an arc of length, then the meaure of, in radian i: r r r r ( in radian) Note: When applying the formula r, the value of mut be in radian. Example A central angle in a circle of radiu 3 cm cut off an arc of length 6 cm. What i the radian meaure of. r 6 cm 3 cm rad 11

2 Example A circle ha radiu 18.0 cm. Find the length of the arc intercepted by a central angle with meaure 3 radian. Given: r 8 3 rad, r 18.0 cm cm 1.44 cm Example The minute hand of a clock i 1. cm long. To two ignificant digit, how far doe the tip of the minute hand move in 0 minute? Given: r = 1. cm One complete rotation = 1 hour = 60 minute = r cm 1

3 Example A peron tanding on the earth notice that a 747 jet flying overhead ubtend an angle If the length of the jet i 30 ft., find it altitude to the nearet thouand feet. r r (180) ,000 ft Example A rope i being wound around a drum with radiu ft. How much rope will be wound around the drum if the drum i rotated through an angle of 39.7? r feet 13

4 Area of a Sector A ector of a circle i a portion of the interior of a circle intercepted by a central angle. A r r r A 1 r Area of ec tor A Central angle Area of circle One full rotation r Definition If (in radian) i a central angle in a circle with radiu r, then the area of the ector formed by an angle i given by A 1 r ( in radian) Example Find the area of the ector formed by a central angle of 1.4 radian in a circle of radiu.1 meter Given: r =.1 m = 1.4 A 1 r 1 (.1) (1.4) 3.1m 14

5 Example If the ector formed by a central angle of 15 ha an area of Given: A 1 r A 3 1 r r 1 r cm, find the radiu of a circle. r 8 r cm Example A lawn prinkler located at the corner of a yard i et to rotate 90 and project water out 30.0 ft. To three ignificant digit, what area of lawn i watered by the prinkler? Given: A 1 r 90 r 30 ft 1 (30) 707 ft 15

6 Exercie Section. Arc Length and Sector Area 1. The minute hand of a clock i 1. cm long. How far doe the tip of the minute hand travel in 40 minute?. Find the radian meaure if angle, if i a central angle in a circle of radiu r = 4 inche, and cut off an arc of length = 1 inche. 3. Give the length of the arc cut off by a central angle of radian in a circle of radiu 4.3 inche 4. A pace huttle 00 mile above the earth i orbiting the earth once every 6 hour. How long, in hour, doe it take the pace huttle to travel 8,400 mile? (Aume the radiu of the earth i 4,000 mile.) Give both the exact value and an approximate value for your anwer. 5. The pendulum on a grandfather clock wing from ide to ide once every econd. If the length of the pendulum i 4 feet and the angle through which it wing i 0. Find the total ditance traveled in 1 minute by the tip of the pendulum on the grandfather clock. 6. Reno, Nevada i due north of Lo Angele. The latitude of Reno i 40, while that of Lo Angele i 34 N. The radiu of Earth i about 4000 mi. Find the north-outh ditance between the two citie. 7. The firt cable railway to make ue of the figure-eight drive ytem wa a Sutter Street Railway. Each drive heave wa 1 feet in diameter. Find the length of cable riding on one of the drive heave. 1 ft 16

7 8. The diameter of a model of George Ferri Ferri wheel i 50 feet, and i the central angle formed a a rider travel from hi or her initial poition to poition. Find the ditance traveled by the rider if = 45 and if =105. P 0 P 1 9. Two gear are adjuted o that the maller gear drive the larger one. If the maller gear rotate through an angle of 5, through how many degree will the larger gear rotate? 10. Two gear are adjuted o that the maller gear drive the larger one. If the maller gear rotate through an angle of 300, through how many degree will the larger rotate? 11. The rotation of the maller wheel caue the larger wheel to rotate. Through how many degree will the larger wheel rotate if the maller one rotate through 60.0? 1. Find the radiu of the larger wheel if the maller wheel rotate 80 when the larger wheel rotate Lo Angele and New York City are approximately,500 mile apart on the urface of the earth. Auming that the radiu of the earth i 4,000 mile, find the radian meaure of the central angle with it vertex at the center of the earth that ha Lo Angele on one ide and New York City in the other ide. 17

8 14. Find the number of regular (tatute) mile in 1 nautical mile to the nearet hundredth of a mile. (Ue 4,000 mile for the radiu of the earth). 15. If two hip are 0 nautical mile apart on the ocean, how many tatute mile apart are they? 16. If a central angle with it vertex at the center of the earth ha a meaure of 1, then the arc on the urface of the earth that i cut off by thi angle (know a the great circle ditance) ha a meaure of 1 nautical mile. 17. How many inche will the weight rie if the pulley i rotated through an angle of 71 50? Through what angle, to the nearet minute, mut the pulley be rotated to raie the weight 6 in? 18. The figure how the chain drive of a bicycle. How far will the bicycle move if the pedal are rotated through 180? Aume the radiu of the bicycle wheel i 13.6 in. 18

9 19. The circular of a Medicine Wheel i 500 yr old. There are 7 aboriginal poke in the wheel, all equally paced. a) Find the meaure of each central angle in degree and in radian. b) The radiu meaure of each of the wheel i 76.0 ft, find the circumference. c) Find the length of each arc intercepted by conecutive pair of poke. d) Find the area of each ector formed by conecutive poke, 0. Find the radiu of the pulley if a rotation of 51.6 raie the weight 11.4 cm. 1. The total arm and blade of a ingle windhield wiper wa 10 in. long and rotated back and forth through an angle of 95. The haded region in the figure i the portion of the windhield cleaned by the 7-in. wiper blade. What i the area of the region cleaned? 19

10 . A frequent problem in urveying city lot and rural land adjacent to curve of highway and railway i that of finding the area when one or more of the boundary line i the arc of the circle. Find the area of the lot. D C A B 3. Nautical mile are ued by hip and airplane. They are different from tatue mile, which equal 580 ft. A nautical mile i defined to be the arc length along the equator intercepted by a central angle AOB of 1 min. If the equatorial radiu i 3963 mi, ue the arc length formula to approximate the number of tatute mile in 1 nautical mile. 4. The ditance to the moon i approximately 38,900 mi. Ue the arc length formula to etimate the diameter d of the moon if angle i meaured to be

11 Section. Arc Length and Sector Area Exercie The minute hand of a clock i 1. cm long. How far doe the tip of the minute hand travel in 40 minute? 40min 40min 60 r (1.) cm 4. 3 rad rad min Exercie Find the radian meaure if angle, if i a central angle in a circle of radiu r = 4 inche, and cut off an arc of length = 1 inche. r rad Exercie Give the length of the arc cut off by a central angle of radian in a circle of radiu 4.3 inche Given: r 4.3() 8.6 in rad, r 4.3 in 7

12 Exercie A pace huttle 00 mile above the earth i orbiting the earth once every 6 hour. How long, in hour, doe it take the pace huttle to travel 8,400 mile? (Aume the radiu of the earth i 4,000 mile.) Give both the exact value and an approximate value for your anwer. r rad rad x hr rad 6 hr (6) x 1.91 hr Exercie The pendulum on a grandfather clock wing from ide to ide once every econd. If the length of the pendulum i 4 feet and the angle through which it wing i 0. Find the total ditance traveled in 1 minute by the tip of the pendulum on the grandfather clock. Since 0 0 rad The length of the pendulum wing in 1 econd: 4 r 4 ft. 9 9 In 60 econd, the total ditance traveled 4 d feet feet. 8

13 Exercie Reno, Nevada i due north of Lo Angele. The latitude of Reno i 40, while that of Lo Angele i 34 N. The radiu of Earth i about 4000 mi. Find the north-outh ditance between the two citie. The central angle between two citie: r mile 30 rad Exercie The firt cable railway to make ue of the figure-eight drive ytem wa a Sutter Street Railway. Each drive heave wa 1 feet in diameter. Find the length of cable riding on one of the drive heave. Since rad The length of the cable riding on one of the drive heave i: r feet, 9

14 Exercie The diameter of a model of George Ferri Ferri wheel i 50 feet, and i the central angle formed a a rider travel from hi or her initial poition Po to poition P1. Find the ditance traveled by the rider if = 45 and if =105. r D 50 For = 45 r ft For = 105 r 15 ft ft Exercie Two gear are adjuted o that the maller gear drive the larger one. If the maller gear rotate through an angle of 5, through how many degree will the larger gear rotate? The motion of the larger gear: The arc length on the maller gear i: r cm The arc length on the larger gear i: r rad 10

15 Exercie If a central angle with it vertex at the center of the earth ha a meaure of 1, then the arc on the urface of the earth that i cut off by thi angle (know a the great circle ditance) ha a meaure of 1 nautical mile rad r mi Exercie If two hip are 0 nautical mile apart on the ocean, how many tatute mile apart are they? rad r

16 Exercie Two gear are adjuted o that the maller gear drive the larger one. If the maller gear rotate through an angle of 300, through how many degree will the larger rotate? Both gear travel the ame arc ditance (), therefore: r r Exercie The rotation of the maller wheel caue the larger wheel to rotate. Through how many degree will the larger wheel rotate if the maller one rotate through 60.0? Both gear travel the ame arc ditance (), therefore: r r Exercie Find the radiu of the larger wheel if the maller wheel rotate 80 when the larger wheel rotate 50. r r 1 1 r r 18.7 cm 50 1

17 Exercie How many inche will the weight rie if the pulley i rotated through an angle of 71 50? Through what angle, to the nearet minute, mut the pulley be rotated to raie the weight 6 in? r in r rad Exercie The figure how the chain drive of a bicycle. How far will the bicycle move if the pedal are rotated through 180? Aume the radiu of the bicycle wheel i 13.6 in. 180 rad The ditance for the pedal gear: r 4.7 in 1 1 For the maller gear: r 1.38 The wheel ditance: r in 13

18 Exercie The circular of a Medicine Wheel i 500 yr old. There are 7 aboriginal poke in the wheel, all equally paced. a) Find the meaure of each central angle in degree and in radian. b) The radiu meaure of each of the wheel i 76.0 ft, find the circumference. c) Find the length of each arc intercepted by conecutive pair of poke. d) Find the area of each ector formed by conecutive poke, a) The central angle: b) 40 rad rad C r 76 ft c) Since r 76 r ft 7 d) Area 1 r ft Exercie Find the radiu of the pulley if a rotation of 51.6 raie the weight 11.4 cm. r cm 14

19 Exercie The total arm and blade of a ingle windhield wiper wa 10 in. long and rotated back and forth through an angle of 95. The haded region in the figure i the portion of the windhield cleaned by the 7-in. wiper blade. What i the area of the region cleaned? The total angle: A 1 A rad : The area of arm only (not cleaned by the blade). : The area of arm and the blade. The total cleaned area: A A A in A A Exercie A frequent problem in urveying city lot and rural land adjacent to curve of highway and railway i that of finding the area when one or more of the boundary line i the arc of the circle. Find the area of the lot. Uing the Pythagorean theorem: AC r Total area = Area of the ector (ADC) + Area of the triangle (ABC) Total area 60 1 r 1 AB BC yd D A C B 15

20 Exercie Nautical mile are ued by hip and airplane. They are different from tatue mile, which equal 580 ft. A nautical mile i defined to be the arc length along the equator intercepted by a central angle AOB of 1 min. If the equatorial radiu i 3963 mi, ue the arc length formula to approximate the number of tatute mile in 1 nautical mile The arc length: rad r There are 1.15 tatute mile in 1 nautical mile. Exercie The ditance to the moon i approximately 38,900 mi. Ue the arc length formula to etimate the diameter d of the moon if angle i meaured to be r mi 16