Please solve the problem in detail

 

The height of the cone is 30cm.A small cone is cut off at the top by a plane parallel to base.If its volume be 1/27 of the volume of the given cone, at what height above the surface the section made?

 

 

Thanks

1cm

Describe the volume of the little cone and large cone in terms of its dimensions (r, height). Add in the variable y which is the height in which you are going to cut it at. Then set the condition:

 

Small cone volume = 1/27 Large cone volume

 

And solve from there.

the big cone volume = (pi)r²30 * 1/3 = (pi)r²10

what is the ratio of its height to volume?

30cm : (pi)r²10 cm³

which is:

3cm : (pi)r² cm³

 

note that the height of the maller cone is directly proportional to its volume.

 

what's the volume of the smaller cone?

big cone volume / 27, right?

therefore, is:

(pi)r²10 cm³ / 27, which is:

(pi)r²(10/27) cm³

 

the height thus will be, according to the ratio of height and volume,

3 / (pi)r² * (pi)r²(10/27) =

3 * 10/27 = 30/27 cm

 

Albert

I don't get this answer. ..........

PrimaryGun, is this the reason you send me a PM with a 'hi'??

20 cm

 

v1,h1,r1=v,h,r smal cone

v2,h2,r2=v,h,r big cone

 

v1=h1*(r1)^2*pi/3

v2=h2*(r2)^2*pi/3

 

h1/h2=r1/r2 (similar triangles)

x (required height)=h2-h1

 

v1/v2=1/27=(h1/h2)^3

=>h1/h2=1/3

 

x=30-10=20

 

Hope that is detailed enough.

Phew ! At least someone got the answer right.

 

But why are folks providing complete (at least one person has) solutions, when the OP has shown no original effort ?

 

To the OP : Please show anything that you have tried or any ideas you have. You won't gain much by having others do your homework for you. Fortunately, there are enough wrong answers here that the correct solution is not yet revealed.

 

What do you think are the steps you must take; and what important properties must you take advantage of ?

 

To Dave/matt : Is this kind of thing the norm here ?

Not usually, no. However, people had already posted complete solutions before I had a chance to survey the thread, so I couldn't really do much about it then.

alright, I am not gonna do people's HW again, just give hints/suggestions. new here.

A small cone is cut off at the top by a plane parallel to base.

From that you know that they are mathmatically similar, the shape is the same, but the scale isn't.

If its volume be 1/27 of the volume of the given cone

If it's volume is 1/27 then it is 1/27 of the larger cone in volume and dimensions and any other mesurement you care to take.

 

That's all you need to know. You can do the number crunching yourself, that should be easy.

Hmm, guess my answer is wrong, for 30- 30/27 cm......

It is more like 30 - 30/(27^1/3) since height is one-dimensional, while volume is 3D. If it is simpler think of a cube. 8 times bigger volume means only twice the side. Same with any figure that is scaled.

I simply set V1 = 27(V2) and got an answer of 1.111, same as albert. That is to say, the cut needs to be made 1.1111 cm below the tip.

PrimaryGun, is this the reason you send me a PM with a 'hi'??

Yes, I got 20cm by concerning the cones are in similar shape.

Well ,

Thanks everybody for your efforts. I now should keep in mind to provide things i know before posting a new thread.

Well the correct answer is 20cm.

 

Thanks