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

- anonymous

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- Shayaan_Mustafa

Hi MertsJ.

- Mertsj

Hi there.

- Shayaan_Mustafa

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## More answers

- Mertsj

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

- Mertsj

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

- Shayaan_Mustafa

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

- Mertsj

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

- Shayaan_Mustafa

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

- Shayaan_Mustafa

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

- Shayaan_Mustafa

let us start.

- Mertsj

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.

- Mertsj

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

- anonymous

Tried to figure out the diagram too

- Mertsj

Is there any information as to what y represents?

- Shayaan_Mustafa

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.

- Shayaan_Mustafa

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

- anonymous

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

- Mertsj

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.

- Shayaan_Mustafa

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

- Mertsj

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?

- anonymous

I think one end of the cylinder is touching each other

- Shayaan_Mustafa

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

- Shayaan_Mustafa

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

- Mertsj

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

- anonymous

Trying to visualize it

- Shayaan_Mustafa

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

- anonymous

It can be mean that the cylinder is in lying position

- Mertsj

|dw:1327783724290:dw|

- Shayaan_Mustafa

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

- Shayaan_Mustafa

@MertsJ.
Are their axes collinear or not?

- Mertsj

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?

- anonymous

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

- Mertsj

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

- anonymous

a-x for one side

- Shayaan_Mustafa

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

- Mertsj

Yes

- Mertsj

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

- Shayaan_Mustafa

a-x for one side
yes it is.

- anonymous

engineering degree

- Mertsj

|dw:1327784324342:dw|

- Mertsj

Is it a calculus problem?

- anonymous

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)

- anonymous

Partial diff

- anonymous

but the answer ain't matchin

- anonymous

what is y???

- Shayaan_Mustafa

yes this y is confusing to us.

- Shayaan_Mustafa

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

- Shayaan_Mustafa

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

- Mertsj

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.

- anonymous

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

- anonymous

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

- Mertsj

|dw:1327784699588:dw|

- Shayaan_Mustafa

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

- Mertsj

Could that be what is meant?

- anonymous

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

- Mertsj

And it also says that is it inscribed.

- Shayaan_Mustafa

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

- Shayaan_Mustafa

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

- Mertsj

|dw:1327784979966:dw|

- Mertsj

Perhaps like that?

- Shayaan_Mustafa

not this too. i am sure.

- anonymous

I think this is the right diagram

- Mertsj

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

- Shayaan_Mustafa

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.

- anonymous

Let try. First take vol of sphere -vol of fist cyl. We got 4 equal outer region that does not touch the cyl

- Shayaan_Mustafa

ok try. i will just see.

- anonymous

Maybe that the y

- Shayaan_Mustafa

will any one try to solve it?

- Mertsj

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

- anonymous

How you get that

- anonymous

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

- Shayaan_Mustafa

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

- Shayaan_Mustafa

yes this y creating confusion. still.

- anonymous

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

- Mertsj

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

- Mertsj

And the total volume of the two cylinders is
\[2\pi x(a ^{2}-x ^{2})+\pi (y+2x)[a ^{2}-(y+x)^{2}]^{2}\]

- Mertsj

Any comment?

- anonymous

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?

- anonymous

That was the question given to me

- anonymous

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

- anonymous

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

- Shayaan_Mustafa

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

- anonymous

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.

- anonymous

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

- anonymous

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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