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cbrsam

  • 2 years ago

How to solve this. A cylinder of length 2x is inscribed in a sphere of radius a . Between one end ot this cylinder & sphere, another cylinder is inscribed with one end on an end of the first cylinder so that the axes of the cylinders are collinear. Show that the sum of the volumes of the two cylinders is, V= 2(Pi) ( x + y )^( a^2 - x^2 - 4y) Thanks

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  1. Shayaan_Mustafa
    • 2 years ago
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    Hi MertsJ.

  2. Mertsj
    • 2 years ago
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    Hi there.

  3. Shayaan_Mustafa
    • 2 years ago
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    Are you understanding question?

  4. Mertsj
    • 2 years ago
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    This is a hard problem. I think we need Satellite.

  5. Mertsj
    • 2 years ago
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    I think maybe. I tried to draw it and that is not easy.

  6. Shayaan_Mustafa
    • 2 years ago
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    nothing is hard. we can do anything. just need of help.

  7. Mertsj
    • 2 years ago
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    Also it would be helpful if we knew what y represents.

  8. Shayaan_Mustafa
    • 2 years ago
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    i think coordinates. because he talk about axes. isn't it?

  9. Shayaan_Mustafa
    • 2 years ago
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    first we need a diagram. otherwise it will take more time to understand.

  10. Shayaan_Mustafa
    • 2 years ago
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    let us start.

  11. Mertsj
    • 2 years ago
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    The axes of the cylinders. So I would assume that means the line down the middle of the cylinder that is perpendicular to both bases.

  12. Mertsj
    • 2 years ago
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    Or perhaps I am wrong in my assumption that these are right circular cylinders.

  13. cbrsam
    • 2 years ago
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    Tried to figure out the diagram too

  14. Mertsj
    • 2 years ago
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    Is there any information as to what y represents?

  15. Shayaan_Mustafa
    • 2 years ago
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    The axes of the cylinders. So I would assume that means the line down the middle of the cylinder that is perpendicular to both bases. Yeah i agree.

  16. Shayaan_Mustafa
    • 2 years ago
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    A cylinder of length 2x is inscribed in a sphere of radius a |dw:1327783252116:dw| Is this right? for the above mentioned line?

  17. cbrsam
    • 2 years ago
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    I think x & y represent the value of r^2

  18. Mertsj
    • 2 years ago
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    So what we need to know now is the radius of the cylinder so that we can find the area of the bases of the cylinder.

  19. Shayaan_Mustafa
    • 2 years ago
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    can you elaborate the wording from the word "between one end of this cylinder?" so that we could figure it out.

  20. Mertsj
    • 2 years ago
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    So is there some relationship between a sphere and the radius of the circle whose center is a-x units from the endpoint of the diameter?

  21. cbrsam
    • 2 years ago
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    I think one end of the cylinder is touching each other

  22. Shayaan_Mustafa
    • 2 years ago
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    @Mertsj. Here is just a sphere not a circle.

  23. Shayaan_Mustafa
    • 2 years ago
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    we need some more help. now i want to answer this question.

  24. Mertsj
    • 2 years ago
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    But if the cylinder is inscribed in a sphere, isn't the base of the cylinder a circle of the sphere?

  25. cbrsam
    • 2 years ago
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    Trying to visualize it

  26. Shayaan_Mustafa
    • 2 years ago
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    hmmm... yes you are right. but does this matter here?

  27. cbrsam
    • 2 years ago
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    It can be mean that the cylinder is in lying position

  28. Mertsj
    • 2 years ago
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    |dw:1327783724290:dw|

  29. Shayaan_Mustafa
    • 2 years ago
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    yes cbrsam you are right. cylinders are in lying position. therefore their axes are collinear otherwise this is not possible.

  30. Shayaan_Mustafa
    • 2 years ago
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    @MertsJ. Are their axes collinear or not?

  31. Mertsj
    • 2 years ago
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    The problem says they are. I need to put another cylinder in my drawing but I am not good at drawing. Do you want to try?

  32. cbrsam
    • 2 years ago
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    If we see only one cylinders . 2a - 2x is the part that protrude out at the end of cylinder

  33. Mertsj
    • 2 years ago
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    Actually a-x since there is an equal piece at both ends of the cylinder.

  34. cbrsam
    • 2 years ago
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    a-x for one side

  35. Shayaan_Mustafa
    • 2 years ago
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    remember guys. axes must be collinear. otherwise we could not be able to find volume.

  36. Mertsj
    • 2 years ago
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    Yes

  37. Mertsj
    • 2 years ago
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    What class are you taking. Is this a calculus problem?

  38. Shayaan_Mustafa
    • 2 years ago
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    a-x for one side yes it is.

  39. cbrsam
    • 2 years ago
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    engineering degree

  40. Mertsj
    • 2 years ago
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    |dw:1327784324342:dw|

  41. Mertsj
    • 2 years ago
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    Is it a calculus problem?

  42. shaan_iitk
    • 2 years ago
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    okk .. I will approach this as .. the big cylinder (with length 2x) would have radius as sqrt (a^2 - x^2) the smaller cylinder would have radius as sqrt(a^2 - (x+y)^2) .. this is assuming y is the length of the smaller cylinder ... then if I solve .. i get sum of volume as pie*(a^2 + y^2 - x^2)*(y + 2x)

  43. cbrsam
    • 2 years ago
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    Partial diff

  44. shaan_iitk
    • 2 years ago
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    but the answer ain't matchin

  45. shaan_iitk
    • 2 years ago
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    what is y???

  46. Shayaan_Mustafa
    • 2 years ago
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    yes this y is confusing to us.

  47. Shayaan_Mustafa
    • 2 years ago
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    @shaan_iitk. I got the same ans. :-(

  48. Shayaan_Mustafa
    • 2 years ago
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    is this question complete? may be he forget to give description about y.

  49. Mertsj
    • 2 years ago
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    Perhaps y is the radius of the small cylinder and it has height 2x since the problem specifies that one end is on the end of the first cylinder.

  50. cbrsam
    • 2 years ago
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    That the question I got nothing else specified. Can it be the second cylinder is the same size

  51. shaan_iitk
    • 2 years ago
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    well then the question is incomplete ..isn't it.. there has to be some meaning of y

  52. Mertsj
    • 2 years ago
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    |dw:1327784699588:dw|

  53. Shayaan_Mustafa
    • 2 years ago
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    really i am an electronics engineer, solid state physicist, semi conductor physicist, interest in cosmology. but not a good mathematician.

  54. Mertsj
    • 2 years ago
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    Could that be what is meant?

  55. cbrsam
    • 2 years ago
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    It stated that the other cylinder only touches one side so it should be outside

  56. Mertsj
    • 2 years ago
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    And it also says that is it inscribed.

  57. Shayaan_Mustafa
    • 2 years ago
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    @Mertsj Read line. "Between one end ot this cylinder & sphere", another cylinder is inscribed cylinder and sphere.

  58. Shayaan_Mustafa
    • 2 years ago
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    we need another help. until we don't figure it. we can't move ahead.

  59. Mertsj
    • 2 years ago
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    |dw:1327784979966:dw|

  60. Mertsj
    • 2 years ago
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    Perhaps like that?

  61. Shayaan_Mustafa
    • 2 years ago
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    not this too. i am sure.

  62. cbrsam
    • 2 years ago
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    I think this is the right diagram

  63. Mertsj
    • 2 years ago
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    So do we agree that the volume of the large cylinder is 2pi(x)(a^2-x^2)?

  64. Shayaan_Mustafa
    • 2 years ago
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    kindly review again this line. Between one end ot this cylinder & sphere, "another cylinder is inscribed" another cylinder means the whole other cylinder. can't see it? huh.

  65. cbrsam
    • 2 years ago
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    Let try. First take vol of sphere -vol of fist cyl. We got 4 equal outer region that does not touch the cyl

  66. Shayaan_Mustafa
    • 2 years ago
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    ok try. i will just see.

  67. cbrsam
    • 2 years ago
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    Maybe that the y

  68. Shayaan_Mustafa
    • 2 years ago
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    will any one try to solve it?

  69. Mertsj
    • 2 years ago
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    \[V=2\pi x(a ^{2}-x ^{2})\] Volume of first cylinder

  70. cbrsam
    • 2 years ago
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    How you get that

  71. Jemurray3
    • 2 years ago
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    It'd be lovely to have a definitive answer of what y is still... :)

  72. Shayaan_Mustafa
    • 2 years ago
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    hmm... may be you are right Mertsj. this seems to be the volume of the first cylinder.

  73. Shayaan_Mustafa
    • 2 years ago
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    yes this y creating confusion. still.

  74. cbrsam
    • 2 years ago
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    What I get is v = 4 Pi. ( a -x )^2

  75. Mertsj
    • 2 years ago
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    Let y be the distance that the small cylinder extends below the large cylinder. Then the height of the small cylinder is y+2x and the radius of the small cylinder is a^2-(y+x)^2 and it's volume is pi(y+2x)[(a^2-(y+x)^2]^2

  76. Mertsj
    • 2 years ago
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    And the total volume of the two cylinders is \[2\pi x(a ^{2}-x ^{2})+\pi (y+2x)[a ^{2}-(y+x)^{2}]^{2}\]

  77. Mertsj
    • 2 years ago
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    Any comment?

  78. Jemurray3
    • 2 years ago
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    Where did this problem come from? I would be willing to nearly bet my life that there is some indication of what y is supposed to be in the problem, variables are never implicitly defined like that.... additionally, could we get a clarification on the correct answer? Because dimensionally what's typed in the box can't be right, so was there a typo, or what?

  79. cbrsam
    • 2 years ago
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    That was the question given to me

  80. Jemurray3
    • 2 years ago
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    In that case, I would ask you to clarify these questions with the source.

  81. accessibm
    • 2 years ago
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    It says the 2 cylinder are collinear. Thus, I think both the drawings are incorrect?

  82. Shayaan_Mustafa
    • 2 years ago
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    yes. i agree with accessibm. That is what i am also saying.

  83. accessibm
    • 2 years ago
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    Volume of first cylinder =2πX(a^2−x^2) this looks fine. if we confirm the vol of 1st cylinder is as above. we can substract from the sum of vol. which gives us the vol of the 2nd cylinder. perform some reverse engineering and hopefully get some ans? so far cannot conclude any ans. maybe there is typo error in the sum of the vol. stated in the question. Do confirm.

  84. cbrsam
    • 2 years ago
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    Hi Mertsj, shayaan, accessibm, Jemurray already check with the source, there is some mistake in the question. The 2nd cylinder is of length 2y is inscribed. The equation is V= 2(Pi) (x+y) (a^2 - x^2 -4y ) Thanks

  85. accessibm
    • 2 years ago
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    Then the vol of the sm cyl will be 2πY(a^2−(X+2Y)^2). add both vol. Still dont not add up to the eqn. Do confirm if there is typo error? I believe the approach is correct. The diagram should look like a urn in a sphere from 2D front view.

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