anonymous 5 years ago Find the area of the shaded region enclosed in a semicircle of diameter 8 centimeters. The length of the chord AB is 6 centimeters. Answer to 2 decimal places. (Hint: Triangle ABC is a right triangle.) put in cm^2 http://www.webassign.net/sat/10-4-29.gif

1. anonymous

Several ways to this, higher math use double integrals polar. However, it could be done geometrically. Calculate the area of the triangle; calculate area of semi-circle; Subtract

2. amistre64

this one right?

3. anonymous

yup

4. amistre64

ah, thats such a pretty picture lol

5. anonymous

haha lol. its 3am and im doin maths lol. thats commiitment haha

6. amistre64

diam = 8; radius = 4 the area of each 'segment' = area sector - area triangle

7. amistre64

lets get the larger angle; using law of cos; in a good way lol 6^2 = 4^2 + 4^2 - 2(4)(4) cos(C) 36 - 16 - 16 ----------- = cosC; C = cos^-1 of that stuff right? -32

8. amistre64

36 - 32 = 4 -4 --- = -1/8 32

9. anonymous

haha looks good

10. anonymous

-1/8 ill giv it a try

11. amistre64

151.04 is what i get in degrees lol

12. amistre64

13. anonymous

hahaha. so im guessin the -1/8 is in cm^2 ?

14. amistre64

..... maybe :)

15. anonymous

hahahaha

16. amistre64

there is a simpler way.... my other personality just was sitting there laughing at me and finally told me this..

17. amistre64

since the radius = 4; what is the area of the half cirlce? pi r^2 ----- right? 2

18. anonymous

righto

19. amistre64

8pi total area of the half....

20. amistre64

subtract area of the tri now

21. anonymous

ohh i see. and area of tri is found by cos formula right?

22. amistre64

or vectors :)

23. amistre64

but yeah

24. anonymous

how do u use vectors

25. amistre64

draw lines with arrows attached

26. amistre64

lets go with law of cosines to figure angles tho

27. anonymous

area triangle (1/2)bh

28. amistre64

to find the height we gotta know an angle

29. anonymous

Height is given, right triangle.

30. amistre64

cos^(-1)[-1/8] gives us an inside angle of what... 115.yada right?

31. amistre64

151.04497562814015224245793040175 i meant lol

32. amistre64

180 - angle ---------- = angle we can use 2

33. anonymous

i got 97.180

34. amistre64

14.477512 is the angle onthe left side

35. amistre64

6 sin(angle) = height

36. anonymous

kinda lost me there

37. amistre64

the interior angle at the center there is abt 151 right?

38. anonymous

39. amistre64

we got an isosoleces triangle so the other 2 angles are the same right? 180 - 151 --------- = angle we need 2

40. amistre64

why do i get dofferent result lol

41. amistre64

it is -1/8 right?

42. anonymous

lol not sure but is it cos(^-1)(-1/8) ?

43. anonymous

yeah,, the -1/8 part cud be the mess up

44. amistre64

6^2 = 4^2+4^2 -2(4)(4) cosC 36 - 32 ------- = 4/-32 = -1/8 -32

45. anonymous

nah it cant be then

46. amistre64

cos^-1(-1/8) = abt. 97

47. anonymous

yes thats the one

48. amistre64

it must be my fat fingers gettin gin the way lol

49. anonymous

hahaha its all good

50. amistre64

180 - 97.180755781458281323038995626149 -------- 2

51. amistre64

41.409622109270859338480502186926 now we got this angle we can determine the height by its sin times that 6 distance

52. amistre64

3.968... is what I get for height

53. amistre64

h*8 --- = area of tri 2

54. anonymous

lol for the height i got -0.92

55. amistre64

15.8745 = area tri now subtract that from the area of the hemicircle

56. amistre64

lol.... do it again

57. anonymous

so u did Sin(41.40*6) for height?

58. amistre64

8pi - 15.8745.... = 9.258....

59. amistre64

the 6 is outside

60. amistre64

6 sin(angle)

61. anonymous

ohhh

62. anonymous

okah so its 9.258 cm^2

63. amistre64

thats what i get; but then again.... i am just an idiot in disguise ;)

64. anonymous

lol as if. u answered so many of my q's!!!

65. anonymous

and wallah its correct!!

66. amistre64

yay!!

67. anonymous

double yay!!!