## iop360 Group Title A large container has the shape of a frustum of a cone with top radius 8m, bottom radius 2m, and height 6m. The container is being filled with water at the constant rate of 2.9m^3/min. At what rate is the level of water rising at the instant the water is 1m deep? Answer: 10.3cm/min one year ago one year ago

1. iop360 Group Title

$\frac{ dV }{ dt } = +2.9, \frac{ dH }{ dt }=?$

2. Chlorophyll Group Title

@iop360 Do you have the Volume formula?

3. iop360 Group Title

of a regular cone, yes

4. iop360 Group Title

$V = \frac{ 1 }{ 3 }(\pi)r^2h$

5. Chlorophyll Group Title

No this one, have 2 bases!

6. iop360 Group Title

i think what we are supposed to do though is make it into two cones though

7. iop360 Group Title

|dw:1353202550485:dw|

8. iop360 Group Title

9. iop360 Group Title

b and B represent the different volumes?

10. iop360 Group Title

how did you derive this formula

11. iop360 Group Title

oh ok

12. iop360 Group Title

ill try it. thanks!

13. iop360 Group Title

hmm

14. iop360 Group Title

have you tried it?

15. iop360 Group Title

16. iop360 Group Title

i got .383...m/min

17. iop360 Group Title

answer is supposed to be 10.3 cm/min

18. iop360 Group Title

oh wait i think i made an error.. let me recalculate

19. iop360 Group Title

my expression for V is: $V = [\frac{ 16 }{ 27 }(\pi)h^3 + \frac{ 4}{ 27 }(\pi)h^3 + \frac{ 1 }{ 9 }(\pi)h^3]$

20. iop360 Group Title

what did you get?

21. iop360 Group Title

oh wait.. 1/27, not 1/9

22. iop360 Group Title

ok yeah thats what i get now

23. iop360 Group Title

taking its derivative, you get $\frac{ 7 }{ 3 }(\pi)h^2$

24. iop360 Group Title

dh/dt beside it of course

25. iop360 Group Title

(2.9)/(7pi/3) = dh/dt after you plug h =1, which does nothing

26. iop360 Group Title

0.395...m^3/min

27. iop360 Group Title

i think we did it right, not sure why its resulting in the wrong answer

28. iop360 Group Title

did you get 0.395...m/min

29. iop360 Group Title

doesnt that still result in 39.5cm/min then?

30. iop360 Group Title
31. iop360 Group Title

it might be a solution, but i dont get it

32. iop360 Group Title

hmm is there anything extra in the formula you may have forgotten

33. iop360 Group Title

34. iop360 Group Title

http://jwilson.coe.uga.edu/emt725/Frustum/Frustum.cone.html this link doesnt have a square root in the formula

35. iop360 Group Title

bleh, the formula without the sq.root doesnt work either...

36. iop360 Group Title

!!! i think i got it

37. iop360 Group Title

forgot the pi from the above forumla

38. iop360 Group Title
39. iop360 Group Title

i got 10.1 though...calculator didnt take exact values

40. iop360 Group Title

$V = \frac{ (\pi)h }{ 3 }[R^2 + Rr + r^2]$

41. iop360 Group Title

im going to calculate it again..

42. iop360 Group Title

im getting 10.05... cm/min that is slightly off

43. iop360 Group Title

hm ok

44. iop360 Group Title

thanks

45. iop360 Group Title

im going to try it with different values

46. iop360 Group Title

oh wait...i used base for r in that last formula

47. iop360 Group Title

uh oh

48. iop360 Group Title

ill try it with r/R

49. iop360 Group Title

annnnnnd were back to 21pi/27 h^3

50. iop360 Group Title

yeah, i accidently used the B/b for that forumla instead of R/r

51. iop360 Group Title

and yet i got close to a right answer

52. iop360 Group Title

do we sub in h = 1 or h =6? h = 1 is right isnt it?

53. Chlorophyll Group Title

Yes, we calculate the instant rate when h = 1

54. iop360 Group Title

the other guy reading this question: are you doing this question?

55. Callisto Group Title

*learning* ._.!

56. iop360 Group Title

i could switch the values for the question and try it again if you want, haha

57. iop360 Group Title

ok i will

58. iop360 Group Title

a large container has the shape of a frustum of a cone with top radius 10m, bottom radius 6m, and height 4m. The container is being filled with water at the constant rate of 2.9m^3/min. At what rate is the level of water rising at the instant rate the water is 2m deep?

59. iop360 Group Title

60. iop360 Group Title

alright

61. iop360 Group Title

i get 1.9 ..hmmm

62. iop360 Group Title

it could be a glitch with the thing, maybe.

63. iop360 Group Title

ok, im pretty sure were doing it right. imma close this down

64. iop360 Group Title

thanks for the help

65. Chlorophyll Group Title

Check with your teacher about the formula! I'll sure will message you if I find out something interesting :)

66. iop360 Group Title

thanks!

67. Chlorophyll Group Title

I'm so sorry that I'm underestimate this Related Rate frustum of Cone! Here's the correct logic: V = π/ 3 ( R² + Rr + r² ) H With r = 2 and R is Radius variable and H is the Height variable -> V = π/ 3 ( R² + 2R+ 4 ) H ( 1 ) From the ratio of right triangle: ( R - 2 ) / ( 8 - 2 ) = H/ 6 --> R = H + 2 (2) Plug (2) into (1): => V = π/ 3 [ ( H +2) + 2( H +2) + 4 ] H = π/ 3 [ H³ + 6H² + 12H ] so V' = π/ 3 [ 3H² + 12H + 12 ] H' At H = 1: V' = 27 π/ 3 * H' 2.9 = 9 π * H' Thus H' = 2.9 / 9π = .1025 m/ min = 10.3 cm/ min

68. iop360 Group Title

thanks!

69. iop360 Group Title

@Chlorophyll one question though.. what do you do exactly during the ratios of the triangle part? eg. say my top radius was 9, bottom was 8, and height was 5 how do you set it up exactly?

70. Chlorophyll Group Title

Ratio R and H: ( R - 8 )/ ( 9-8) = H/5 -> R = ( H/5 ) + 8