## cbrsam Group Title 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 2 years ago 2 years ago

1. Shayaan_Mustafa Group Title

Hi MertsJ.

2. Mertsj Group Title

Hi there.

3. Shayaan_Mustafa Group Title

Are you understanding question?

4. Mertsj Group Title

This is a hard problem. I think we need Satellite.

5. Mertsj Group Title

I think maybe. I tried to draw it and that is not easy.

6. Shayaan_Mustafa Group Title

nothing is hard. we can do anything. just need of help.

7. Mertsj Group Title

Also it would be helpful if we knew what y represents.

8. Shayaan_Mustafa Group Title

i think coordinates. because he talk about axes. isn't it?

9. Shayaan_Mustafa Group Title

first we need a diagram. otherwise it will take more time to understand.

10. Shayaan_Mustafa Group Title

let us start.

11. Mertsj Group Title

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 Group Title

Or perhaps I am wrong in my assumption that these are right circular cylinders.

13. cbrsam Group Title

Tried to figure out the diagram too

14. Mertsj Group Title

Is there any information as to what y represents?

15. Shayaan_Mustafa Group Title

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 Group Title

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 Group Title

I think x & y represent the value of r^2

18. Mertsj Group Title

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 Group Title

can you elaborate the wording from the word "between one end of this cylinder?" so that we could figure it out.

20. Mertsj Group Title

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 Group Title

I think one end of the cylinder is touching each other

22. Shayaan_Mustafa Group Title

@Mertsj. Here is just a sphere not a circle.

23. Shayaan_Mustafa Group Title

we need some more help. now i want to answer this question.

24. Mertsj Group Title

But if the cylinder is inscribed in a sphere, isn't the base of the cylinder a circle of the sphere?

25. cbrsam Group Title

Trying to visualize it

26. Shayaan_Mustafa Group Title

hmmm... yes you are right. but does this matter here?

27. cbrsam Group Title

It can be mean that the cylinder is in lying position

28. Mertsj Group Title

|dw:1327783724290:dw|

29. Shayaan_Mustafa Group Title

yes cbrsam you are right. cylinders are in lying position. therefore their axes are collinear otherwise this is not possible.

30. Shayaan_Mustafa Group Title

@MertsJ. Are their axes collinear or not?

31. Mertsj Group Title

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 Group Title

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

33. Mertsj Group Title

Actually a-x since there is an equal piece at both ends of the cylinder.

34. cbrsam Group Title

a-x for one side

35. Shayaan_Mustafa Group Title

remember guys. axes must be collinear. otherwise we could not be able to find volume.

36. Mertsj Group Title

Yes

37. Mertsj Group Title

What class are you taking. Is this a calculus problem?

38. Shayaan_Mustafa Group Title

a-x for one side yes it is.

39. cbrsam Group Title

engineering degree

40. Mertsj Group Title

|dw:1327784324342:dw|

41. Mertsj Group Title

Is it a calculus problem?

42. shaan_iitk Group Title

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 Group Title

Partial diff

44. shaan_iitk Group Title

45. shaan_iitk Group Title

what is y???

46. Shayaan_Mustafa Group Title

yes this y is confusing to us.

47. Shayaan_Mustafa Group Title

@shaan_iitk. I got the same ans. :-(

48. Shayaan_Mustafa Group Title

is this question complete? may be he forget to give description about y.

49. Mertsj Group Title

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 Group Title

That the question I got nothing else specified. Can it be the second cylinder is the same size

51. shaan_iitk Group Title

well then the question is incomplete ..isn't it.. there has to be some meaning of y

52. Mertsj Group Title

|dw:1327784699588:dw|

53. Shayaan_Mustafa Group Title

really i am an electronics engineer, solid state physicist, semi conductor physicist, interest in cosmology. but not a good mathematician.

54. Mertsj Group Title

Could that be what is meant?

55. cbrsam Group Title

It stated that the other cylinder only touches one side so it should be outside

56. Mertsj Group Title

And it also says that is it inscribed.

57. Shayaan_Mustafa Group Title

@Mertsj Read line. "Between one end ot this cylinder & sphere", another cylinder is inscribed cylinder and sphere.

58. Shayaan_Mustafa Group Title

we need another help. until we don't figure it. we can't move ahead.

59. Mertsj Group Title

|dw:1327784979966:dw|

60. Mertsj Group Title

Perhaps like that?

61. Shayaan_Mustafa Group Title

not this too. i am sure.

62. cbrsam Group Title

I think this is the right diagram

63. Mertsj Group Title

So do we agree that the volume of the large cylinder is 2pi(x)(a^2-x^2)?

64. Shayaan_Mustafa Group Title

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 Group Title

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 Group Title

ok try. i will just see.

67. cbrsam Group Title

Maybe that the y

68. Shayaan_Mustafa Group Title

will any one try to solve it?

69. Mertsj Group Title

$V=2\pi x(a ^{2}-x ^{2})$ Volume of first cylinder

70. cbrsam Group Title

How you get that

71. Jemurray3 Group Title

It'd be lovely to have a definitive answer of what y is still... :)

72. Shayaan_Mustafa Group Title

hmm... may be you are right Mertsj. this seems to be the volume of the first cylinder.

73. Shayaan_Mustafa Group Title

yes this y creating confusion. still.

74. cbrsam Group Title

What I get is v = 4 Pi. ( a -x )^2

75. Mertsj Group Title

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 Group Title

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 Group Title

Any comment?

78. Jemurray3 Group Title

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 Group Title

That was the question given to me

80. Jemurray3 Group Title

In that case, I would ask you to clarify these questions with the source.

81. accessibm Group Title

It says the 2 cylinder are collinear. Thus, I think both the drawings are incorrect?

82. Shayaan_Mustafa Group Title

yes. i agree with accessibm. That is what i am also saying.

83. accessibm Group Title

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 Group Title

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 Group Title

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.