Change in volume given the ratio of the heights

[greg_rack]

Homework Statement

Statement attached below

Relevant Equations

$V=\frac{1}{3}\pi h(r_{1}^2+r_{1}r_{2}+r_{2}^2)$

The question is: how do I know the increase in the minor and greater radius, given just the ratio of the heights?

 

[haruspex]

Homework Statement:: Statement attached below
Relevant Equations:: $V=\frac{1}{3}\pi h(r_{1}^2+r_{1}r_{2}+r_{2}^2)$

View attachment 271528
The question is: how do I know the increase in the minor and greater radius, given just the ratio of the heights?

You are told that the shapes are "similar" in the mathematical sense, i.e. all distances change in proportion.

 

[greg_rack]

You are told that the shapes are "similar" in the mathematical sense, i.e. all distances change in proportion.

Got it, thanks!
But the thing is, I would have never remembered the formula of the volume for such a cone... is there a way to find the volume after the transformation without knowing the exact formula with all radiuses and stuff, only with heights?
And also, why does the statement speaks about the heights of the cups?

 

[haruspex]

is there a way to find the volume after the transformation without knowing the exact formula with all radiuses and stuff

Yes. If two objects are mathematically similar, all corresponding distances are in the same ratio, r:1 say. Then all corresponding areas are in the ratio r²:1 and all corresponding volumes are in the ratio r³:1.