Math Is Fun Forum

Au101

Member

Registered: 2010-12-01

Posts: 353

Cones

Hey, me again.

I can't decide whether this is something I've never done, or something I've forgotten, so let's just dive straight in with the question.

22. A plane intersects a right-circular cone and is parallel to the base of the cone. If V is the vertex of the cone and VA the altitude, then let B be the point where the plane cuts VA. It it given that VB:BA = 3:2. Are the following statements true or false?
(i) The ratio of the volume of the small cone (the upper part of the section) to the volume of the whole cone is 8:27.
(ii) The ratio of the surface area of the small cone to the surface area of the whole cone is 9:25.
(iii) The ratio of the volume of the small cone to the volume of the frustum of the cone (the lower part of the section) is 27:125.

Now, I dare not even try to draw this, but I'm pretty confident I do know what the question means. You've got your standard cone, where the "top" (here, I believe, referred to as the 'vertex', though I believe 'apex' is also used) of the cone is directly above the centre of the base. The top of the cone is labelled V and we extend an imaginary line from the top of the cone to the centre of the base, which we'll call A. This line is VA - the "altitude" (i.e. the height of the cone, i.e. the distance from the base of the cone to the point at the top). Now, at a point B along this line a plane cuts the cone into two parts. So we're left with a "small cone" and a "frustum". Now, the ratio of the height of the small cone (VB) to the height of the frustum (BA) is 3:2. This means, then, that the height of the small cone (VB) is 3/5 that of the whole cone (VA) - am I right?

Now, what else do we know. Well, we know the volume of a cone is:

Where r is the radius of the base and h the height of the cone. So we know that for the whole cone h is VA and we know that for the small cone is 3/5VA.

What I don't know is what either of the radii are and I don't know how to work out in what proportion the radii are, which I imagine is important.

I'm also not really sure how to proceed on the basis of this information. What I jotted down is this:

Which I can kind of see makes (i) false (book says it is indeed false), but I don't really understand why - I couldn't explain it to anybody if they asked. I then looked at the next one and, yeah, apparently it's true, but I don't really know why it's true. One of the things that's stalling me is the fact that I'm not sure whether I'm supposed to worry about the radii and - if I am - how I can work out what ratio they're in

Bob

Administrator

Registered: 2010-06-20

Posts: 10,149

Re: Cones

hi Au101,

There's an underlying principle here that will enable you to do the whole question, so let's start with that.

Imagine you have a cube with dimensions a by a by a and another larger cube, with dimensions b by b by b.

The ratio of lengths is a:b

The ratio of surface areas is 6a^2 : 6b^2 = a^2:b^2

The ratio of volumes is a^3 : b^3

Now consider a cuboid 1 by 2 by 3 and an exact enlargement 2 by 4 by 6

The ratio of lengths is 1:2

The area of one of the faces of the small is 2 x 3 = 6 and the corresponding area on the large is 4 x 6 = 24, so the ratio of these areas is 6:24 = 1:4.

You can check other faces for yourself and you'll find that corresponding areas are always 1:4.

The ratio of volumes is 1x2x3 : 2x4x6 = 6 : 48 = 1 : 8

So you can see that the rule for cubes also works for cuboids.

In fact, it is true for any 'similar' solids ie. where one is an exact enlargement of the other.

Why?  Let's consider a sphere.  You can work out it's area by considering it is divided into small squares.  As the surface is curved this will only be approximate, but, by taking smaller and smaller squares, you can get closer and closer to an accurate answer.  As the ratio rule applies to squares it must apply to the area of the sphere.

Similarly, the volume can be found by fitting tiny cubes into the sphere.  As those cubes obey the ratio rule, so will the volume of the sphere.

You can also use the formulas for area and volume to check this out for any ratio.

And this argument can be applied to any solid.  As long as solid 2 is an exact enlargement of solid 1, then:

ratio of lengths is a:b                 ratio of areas is a^2 : b^2               ratio of volumes is a^3 to b^3

Back to your problem.  So you don't have to know the radii; you can say the r1:r2 = 3:5 and areas will be 9:25 and volumes will be 27:125.

But, beware.  The rule only applies to similar solids.  You can use it for ratios in the small cone to the large cone.  You cannot apply it directly to the frustrum, because it isn't the same shape.  But you can still work things out for the frustrum volume like this:

Say the volume of the small cone = 27k where k is some unknown value.  Then the volume of the large cone = 125k.  So the volume of the frustrum is 98k.  When you write down the ratio the k values will cancel, so you don't need to know what it is.

The area is harder as you cannot do a similar subtraction.  The frustrum has a top circle and a bottom circle and this complicates matters.  But you should still be able to say if the statement is true or false.

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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Cones

What a lovely, comprehensive explanation, thanks!