# Thread: Area and Volume Ratio's Problem

1. ## Area and Volume Ratio's Problem

Hey dudes. Sweet forum! Sick place to get help from

I understand it, but like, I just don't understand it fully.
If anyone could run me through it that would be awesome.

Another problem I have is this question:
"If the sides of a cube have doubled, how has the volume changed"

I don't really understand it. If any maths wiz's here could help me out on these things, that would be absolutely awesome.
Cheers

2. Originally Posted by Bender92
Hey dudes. Sweet forum! Sick place to get help from

I understand it, but like, I just don't understand it fully.
If anyone could run me through it that would be awesome.

Another problem I have is this question:
"If the sides of a cube have doubled, how has the volume changed"

I don't really understand it. If any maths wiz's here could help me out on these things, that would be absolutely awesome.
Cheers
Let the sides of a cube be of length $\displaystyle x$. Then the volume is of length $\displaystyle x^3$. Now, if the side length is doubled, then $\displaystyle x$ becomes $\displaystyle 2x$. Now substituting that into the volume formula, the new volume is $\displaystyle (2x)^3=8x^3$.

In general, increasing a side length by factor $\displaystyle a$ will mean that the new side length is $\displaystyle ax$ and the new volume is therefore $\displaystyle (ax)^3=a^3\times x^3$.

Or in other words, if the side length of a cube is increased by factor $\displaystyle a$, its volume is increased by factor $\displaystyle a^3$. This in fact holds for any 3D object (as long as ALL sides are increased by factor $\displaystyle a$).

Similarly, for any plane (2 dimensional) object, increasing the side length by factor $\displaystyle a$ means the area increases by factor $\displaystyle a^2$ (as long as ALL sides are increased by factor $\displaystyle a$). You can prove this yourself