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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: 7,273

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

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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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

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: 7,273

Thanks Stefy.

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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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

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: 20

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: 7,273

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

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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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,131

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 (2015-07-18 10:09:14)*

"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: 7,273

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

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

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,273

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

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

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**championmathgirl****Member**- Registered: 2015-06-01
- Posts: 20

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: 4,131

Hi;

In the image I added DM to create right-angled triangle DMA (for which the lengths of DA and DM are known) in order to obtain the length of AM (by Pythagoras).

*Last edited by phrontister (2015-07-22 20:15:55)*

"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: 7,273

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

Bob

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

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**RandomPieKevin****Member**- Registered: 2015-07-02
- Posts: 29

championmathgirl wrote:

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.

I don't know which base you guys are talking about. Also, is x the radius of the inscribed circle?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 98,268

phrontister wrote:

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

Here is one way:

```
FindInstance[
y/(10 + y) == x/(10 - x) == Sqrt[-x^2 + y^2]/(
6 + Sqrt[-x^2 + y^2]), {x, y}, Reals, 5]
```

And then a tiny ansatz is required.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.** **Thinking is cheating.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,273

RandomPieKevin wrote:

I don't know which base you guys are talking about. Also, is x the radius of the inscribed circle?

Don't forget this is a frustum. That is, you start with a big cone, chop off a small cone at the top, and the frustum is what remains. In post 2 I have made a diagram that shows a vertical cross section through the middle of the solids. The red outline is the frustum. It has a circular top radius x and a circular base, radius to be found.

The sphere fits inside the frustum and touches the top and bottom circles and also the sides. The sphere appears in the diagram as a circle because the cross section cuts through the middle of the sphere. It has a radius of 3.

The sloping side of the frustum is 10.

If a pair of tangents are drawn from a point to a circle they will have equal length. To prove this draw a line from the point to the centre of the circle, making two triangles. It is fairly straight forward to show that these triangles are congruent.

So the top radius, x, is repeated down the sloping side, and hence the lower distance is 10-x. By the same tangent length rule this distance is repeated across the bottom. So the base radius is 10-x.

championmathgirl's method is to drop a perpendicular from the top radius down to the bottom making a right angled triangle. It has sides of 10, 10-x-x, and 6. You can use Pythagoras to work out x.

I like to work out the volume of a frustum by calculating the volumes of the large and small cones, and then subtracting.

Bob

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

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**RandomPieKevin****Member**- Registered: 2015-07-02
- Posts: 29

Oh, so the cross-section is vertical? And that tiny triangle is the top part of the cone?____________________________________________________________________________________________________________________________________

How did you get the base as 20-2x?

____________________________________________________________________________________________________________________________________How do you know that JD=x?

*Last edited by RandomPieKevin (2015-07-21 05:44:33)*

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,273

Yes and yes.

Bob

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

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**RandomPieKevin****Member**- Registered: 2015-07-02
- Posts: 29

How did you get the base as 20-2x?

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**RandomPieKevin****Member**- Registered: 2015-07-02
- Posts: 29

Wait, I got 20pi... It's wrong! I got 1/4\pi as the volume of the top and 81/4\pi as the volume of the whole thing.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,273

RandomPieKevin wrote:

How do you know that JD=x?

From a point you can make two tangents to a circle. The lengths of these two tangents (from point to circle) are equal.

RandomPieKevin wrote:

How did you get the base as 20-2x?

Using the same rule AJ = AH = 10 - x so the base is AB = 20 - 2x

In triangle DAM, DM = FH = 6 and DA = 10, therefore AM = 8.

AH = 10 - x and also x + 8 => 10 - x = x + 8 => x = 1

Therefore AH = 9.

So the large cone is 9x as big as the small cone.

If the height of the small cone = h, then 9 times small height = large height => 9h = 6 + h => h = 3/4

So small cone

and large cone

Bob

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

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