Cone mensuration sum

Question

I am a tenth grade student. I am normally good in mensuration, but a question stumped me.

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

What I am guessing is that we have to apply theorem of similarity here. I did a few steps, but I got confused thereafter. By similarity,

r1/r2=h1/h2,

Where r1 = radius of top cone, r2 = radius of original cone, h1 = height of cut off portion, h2 = 30 cm. But what do we do next?

P.S Sorry for bad LaTeX skills.

Answer 1

Ok, you know that the volume of the original cone is $\frac{\pi}{3} h_1 r_1^2$, and the volume of the second cone is $\frac{\pi}{3} h_2 r_2^2$, so you know that: $$27 = \frac{\frac{\pi}{3} h_1 r_1^2}{\frac{\pi}{3} h_2 r_2^2}$$ But by the result you obtained, this tells us that $$27 = \left( \frac{h_1}{h_2} \right)^3$$

Does this get you there?

Answer 2

Generally, if you scale a body linearly in all directions by a factor $a$, its volume scales as $a^3$. It doesn't matter if the body is shaped like a cube, a sphere, a cone or a Spongebob doll; all that matters is that the scaling is applied in all directions (you can try this out for various shapes for whose volumes you have a formula). In this case, the volume scales as $1/27$, so how must the height (or radius, or any other linear dimension) scale?