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**!nval!d_us3rnam3****Member**- Registered: 2017-03-18
- Posts: 25

A sphere with radius 3 is inscribed in a conical frustum of slant height 10. (The sphere is tangent to both bases and the side of the frustum.) Find the volume of the frustum.

I haven't tried much, I'm just really stuck, can you give me a solution skeleton (i.e. where I prove a few things myself)? Thanks in advance!

~ !nval!d_us3rnam3

P.S. Please, don't give me another link to somewhere else on the website, because I've probably read it already. Just help me out here. Thanks again

P.S.S. If you have time, what's the list of working emojis on this website, other than , , , , , , and ? Thanks yet again!

P.S.S.S. How do people get the title of "Real Member"?

*Last edited by !nval!d_us3rnam3 (2017-04-06 04:02:18)*

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,386

hi !nval!d_us3rnam3

If you 'drop' a perpendicular from A to the base you can work out FB using Pythagoras.

Then use similar triangles to work out EF and GH.

Then it should be straight forward to finish the task.

You'll find a long list of smilies here:

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Real members are appointed by MIF. There is also a set of imaginary members but you'll never see them as they are imaginary.

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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
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What do you mean by imaginary members???

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**phrontister****Real Member**- From: The Land of Tomorrow
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**phrontister****Real Member**- From: The Land of Tomorrow
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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,386

hi iamaditya

It was a joke. Mathematicians call numbers like Square Root (-1) imaginary numbers. There's a whole branch of mathematics devoted to the study, properties and applications of such numbers. Look here for a start:

http://www.mathsisfun.com/numbers/complex-numbers.html

They are called 'imaginary' because the imaginary number line is the image of the real number line. This choice has unfortunately led to some ignorant folk laughing at mathematicians for believing in something that is imaginary. When Isaac Asimov was ridiculed by one such critic, saying how can you have an imaginary number, he replied "Here's a piece of chalk; now hand me a half a piece of chalk." His point was that all numbers have no actual existence; they are all just abstract concepts. Take the number 3 for example. You cannot have a three; you cannot hold it; you cannot see it. All you see is the symbol we use to denote it. Out of context, it has no existence of its own. I have three children; but I have no three.

Thank you Phro for your jokes.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**!nval!d_us3rnam3****Member**- Registered: 2017-03-18
- Posts: 25

bob bundy wrote:

hi !nval!d_us3rnam3

http://i.imgur.com/UBcgXOV.gif

If you 'drop' a perpendicular from A to the base you can work out FB using Pythagoras.

Then use similar triangles to work out EF and GH.

Then it should be straight forward to finish the task.

Well, I'll need to find the volume of the top cone. Either that, or I prove a rather confusing frustum volume formula. I know the height is 6, and FB is 8, and slant height is 10, so now what?

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,386

There are several similar triangles: GHA, GEB, AFB for example. If you call GH = x, and HA = y, you should be able to calculate these by creating and solving simultaneous equations in x and y.

Then you'll have the information you need to calculate the volume of the top cone, the larger cone and hence the frustum.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**!nval!d_us3rnam3****Member**- Registered: 2017-03-18
- Posts: 25

Is there an easier way than that, because I'm still confused? Technically, all we need to do is find the top length of the frustum, then we can find the bottom length of the frustum easily. All that we need after that is to find a formula for the volume of a frustum and prove it. (That doesn't really seem easy, but now I'm desperate. I've visited 6 help sites and I'm still none the wiser. Can you be a little more detailed?)

*Last edited by !nval!d_us3rnam3 (2017-04-06 03:15:22)*

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,386

Let GH = x and HA = y

We know AF/FB = 6/8 so x/y = 6/8 = 3/4 and in triangle GEB, GE/EB = (x+6)/(10-y) = 6/8 = 3/4 as well.

EB = 10-y because EB = DB (equal tangents) and DB = 10 - AD and AD = AH = y (equal tangents again)

So you end up with

4x = 3y and

4x + 24 = 10 - 3y

Those are fairly easy to solve and then you can do the volumes. The frustum formula is just (volume of large cone) minus (volume of small cone).

Bob

Is there an easier way?

I think any method has got to make use of the equal tangents property otherwise the problem would be solvable even when the top, bottom and sides don't make tangents .

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**!nval!d_us3rnam3****Member**- Registered: 2017-03-18
- Posts: 25

I get a negative fraction as y. X is equal to y too, for some reason.... Did you mess up somewhere?

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,386

Please post all your working.

B

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**!nval!d_us3rnam3****Member**- Registered: 2017-03-18
- Posts: 25

bob bundy wrote:

So you end up with

4x = 3y and

4x + 24 = 10 - 3y

Replacing 4x in the second equation with 3y (since 4x = 3y) and adding 3y to both sides gives 6y + 24 = 10. Subtracting 24 from both sides gives 6y = -14, or y=-7/3. We can't have negative lengths. I think the equation screwed up somewhere.

EDIT: Otherwise, your solution is nice and clear, and I feel that I can take the rest.

*Last edited by !nval!d_us3rnam3 (2017-04-06 05:46:23)*

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,386

Sorry, you're right. I did mess up. There's a typo there. On my paper version I had

4x = 3y and

4x + 24 = 30 - 3y.

Hopefully that will lead you correctly to a solution.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**!nval!d_us3rnam3****Member**- Registered: 2017-03-18
- Posts: 25

Thank you so much!

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

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**!nval!d_us3rnam3****Member**- Registered: 2017-03-18
- Posts: 25

What did you get as your cone volumes and final answer? Just checking my answers.

*Last edited by !nval!d_us3rnam3 (2017-04-06 08:55:09)*

"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi !nval!d_us3rnam3;

If you are okay and understand the method Bob gave you then go here for other ideas:

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
- Posts: 766

bob bundy wrote:

hi iamaditya

It was a joke. Mathematicians call numbers like Square Root (-1) imaginary numbers. There's a whole branch of mathematics devoted to the study, properties and applications of such numbers. Look here for a start:

http://www.mathsisfun.com/numbers/complex-numbers.html

They are called 'imaginary' because the imaginary number line is the image of the real number line. This choice has unfortunately led to some ignorant folk laughing at mathematicians for believing in something that is imaginary. When Isaac Asimov was ridiculed by one such critic, saying how can you have an imaginary number, he replied "Here's a piece of chalk; now hand me a half a piece of chalk." His point was that all numbers have no actual existence; they are all just abstract concepts. Take the number 3 for example. You cannot have a three; you cannot hold it; you cannot see it. All you see is the symbol we use to denote it. Out of context, it has no existence of its own. I have three children; but I have no three.

Thank you Phro for your jokes.

Bob

Hi Bob,

I know all about imaginary nos. but I did not get what you meant by Imaginary members. Hmm, now I understand what you meant.

Practice makes a man perfect.

There is no substitute to hard work

All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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