## iop360 3 years ago 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

1. iop360

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

2. Chlorophyll

@iop360 Do you have the Volume formula?

3. iop360

of a regular cone, yes

4. iop360

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

5. Chlorophyll

No this one, have 2 bases!

6. iop360

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

7. iop360

|dw:1353202550485:dw|

8. iop360

9. iop360

b and B represent the different volumes?

10. iop360

how did you derive this formula

11. iop360

oh ok

12. iop360

ill try it. thanks!

13. iop360

hmm

14. iop360

have you tried it?

15. iop360

16. iop360

i got .383...m/min

17. iop360

answer is supposed to be 10.3 cm/min

18. iop360

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

19. iop360

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

what did you get?

21. iop360

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

22. iop360

ok yeah thats what i get now

23. iop360

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

24. iop360

dh/dt beside it of course

25. iop360

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

26. iop360

0.395...m^3/min

27. iop360

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

28. iop360

did you get 0.395...m/min

29. iop360

doesnt that still result in 39.5cm/min then?

30. iop360
31. iop360

it might be a solution, but i dont get it

32. iop360

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

33. iop360

34. iop360

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

35. iop360

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

36. iop360

!!! i think i got it

37. iop360

forgot the pi from the above forumla

38. iop360
39. iop360

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

40. iop360

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

41. iop360

im going to calculate it again..

42. iop360

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

43. iop360

hm ok

44. iop360

thanks

45. iop360

im going to try it with different values

46. iop360

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

47. iop360

uh oh

48. iop360

ill try it with r/R

49. iop360

annnnnnd were back to 21pi/27 h^3

50. iop360

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

51. iop360

and yet i got close to a right answer

52. iop360

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

53. Chlorophyll

Yes, we calculate the instant rate when h = 1

54. iop360

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

55. Callisto

*learning* ._.!

56. iop360

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

57. iop360

ok i will

58. iop360

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

60. iop360

alright

61. iop360

i get 1.9 ..hmmm

62. iop360

it could be a glitch with the thing, maybe.

63. iop360

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

64. iop360

thanks for the help

65. Chlorophyll

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

66. iop360

thanks!

67. Chlorophyll

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

thanks!

69. iop360

@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

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