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**jjoy****Guest**

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.

**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,020

hi jjoy

Welcome to the forum.

The algebra is messy but it should work. Sorry, I cannot find a simpler method.

Two tangents to a circle from a point will be equal.

Then the small (removed) cone will be in the same proportion as the large cone. Hence:

I don't have a simple solution to these, but someone else on the forum will probably put the equations through a computer for us and come up with values for x and y.

Then it should be easy (?) to compute the volume.

Bob

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,922

Hi jjoy and Bob

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,020

Thanks Stefy.

Bob

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,922

No problem.

The volume is now easy to compute.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**championmathgirl****Member**- Registered: 2015-06-01
- Posts: 8

I got 182pi, is that right?

Girls can be just as good as boys at math. We just need to get the same encouragement.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,020

hi championmathgirl

it is not what I have just calculated. As y is not a whole number, I would not expect the volume to be either.

What did you get for the height and radius of the two cones (starting large cone and the one that's cut away)?

Bob

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

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,081

Hi Bob;

I think that championmathgirl's answer is correct.

I understand how you got your equations, but haven't worked out yet how stefy arrived at his solutions for x and y. However, they check out in Geogebra.

In the volume formula below I used the following (borrowing some info from your drawing):

The formula for the volume of a conical frustum is:

So...

*Last edited by phrontister (2015-06-30 04:28:57)*

"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****Moderator**- Registered: 2010-06-20
- Posts: 7,020

hi phrontister

Thank you. I decided to work it out on paper this time, and I get 182pi as well. Whoops!

I shall see if I can get an algebraic solution to the equations as a penance.

Bob

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,020

substitute into the third expression (I'll make an equation later):

simplifying and equating to the 'y' expression:

Rejecting the second solution as y cannot be negative in this problem, we have y = 5/4

Bob

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**championmathgirl****Member**- Registered: 2015-06-01
- Posts: 8

I used a simpler algebra way to find x. Because you could use Pythagorean Theorem and write the base as 16+2x. Then since we also have the base as 20-2x. Setting them equal to each other we get x=1. Then use the ratios y/(y+10)=x(10-x) to find y.

Girls can be just as good as boys at math. We just need to get the same encouragement.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,081

Hi Bob,

I added DM to create right-angled triangle DMA (for which there are 2 known sides) to obtain the length of AM, and thus the value of r in the next step.

*Last edited by phrontister (2015-07-01 11:13:36)*

"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****Moderator**- Registered: 2010-06-20
- Posts: 7,020

championmathgirl. You are indeed a champion. That's a much better way. I award you this:

Bob

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