Uhg all you math people...help!?

Question

I have a cube thats 1,1,1 and one thats 2,2,2 and one thats 3,3,3

Describe what happens to the cube when the volume if doubled tripled or quadrupled and so on?

I need help in finding the pattern the volume of the first cube is 1 and the volume of the second cube is eight and the third 27??


Any help is appreciated :)

I was also wondering if there was a certain thing the surface area was going up by? and is there a certain rate that volume goes up by like *4 each time that kind of thing?

Answer 1

The volume of a cube is x^3, where x is the edge length. Let's say you have a cube whose edge length is x, and a second cube that's doube the voulme. What is the edge length of this second cube? Call it y. This means: 2x^3 = y^3, so y = x*(2^(1/3)), or x times the cube root of 2.

Here's a more general solution. Let's say that you have the "x" cube, and you want to find a new cube whose volume is this volume times a factor of "k". Then kx^3 = y^3, and you end up with y = x*(k^(1/3)). So for example, the measure of the cube's edge whose voume is 3 times that of a 4x4x4 cube is 4*(3^(1/3)).

In short, if you have a cube of a certain edge length and want to build a cube whose volume is k times as big, then you need a new cube whose edge length is (k^(1/3)) times the edge length of the first cube.

EDIT: I just noticed that you added a question about surface area. The surface area of a cube of edge length x is going to be 6x^2 (each side is x^2, and you have 6 sides). If you want the surface area of your new cube to go up by a factor of k, you want k*6x^2 = 6y^2, so y = x*sqrt(k).

For a cube, the volume-to-surface area ratio is x^3 / 6x^2 = x/6. So for every unit the surface area goes up, the volume is going to go up by a factor of x/6

Answer 2

The volume increases in respect to the sides of a cube cubed. For example if you double the length of the sides the the volume is multiplied by 8, and if you triple the sides then the volume increases 27 times.

If you have a 2,2,2 cube and double that to a 4,4,4 cube, the volume is multiplied by 8 and increases from 8 to 64.

If you have a 3,3,3 cube and triple the sides to 9,9,9 the volume is multiplied by 27 and increases to 729.

Answer 3

Volume is length * width * height.

So the example with the size of one, the unit cube. If you double the edge, then the length is twice, height is twice, width is twice.

2 * 2 *2 = 8. So, if a cube's edge is X times larger, the volume is X^3 Raising something to the 3rd power is called "cubing" and now you know why.

Answer 4

well, if it's a cube, then V=s^3 (s being the length of one side)
so when we have x*V, the side length would change by a factor of the cube-root of x

So although the side lengths are increasing linearly, the volumes are increasing at an increasing rate