prove that the ..................................................................?

Question

..volume, total surface area , total length of the edges and the number of vertexes of a solid cube of side x unit is always a perfect cube.....? please help..

Answer 1

NEW ANSWER:

Aha! I think I've figured out what you mean.

I think you mean show that

(Volume) + (Total Surface Area) + (Total Length of Edges) + (Number of Vertices) = A Perfect Cube

If that's the case, as in my original answer below, you have

(x^3) + (6x^2) + (12x) + (8)

You can show (I used the rational roots theorem) that this factors to

(x + 2)^3

This is a cube of (x+2), so it is a perfect cube as long as x is an integer.


Original Answer:

Well, the volume is (length)*(width)*(height). Since these are all x,

V = x^3, so as long as x is an integer, volume is a perfect cube.

Surface area: find the area of a single side, which is (length)*(width). Then multiply by the number of sides, which is 6.

SA = 6x^2. This is a perfect cube if and only if the cube root is an integer, but this doesn't happen for all integers.

There are 12 total edges, each of length x, so the

Total Edge Length = 12x. This is a perfect cube if and only if the cube root of 12x is an integer, but this also doesn't happen for all integers.

There are 8 vertices, and 8 is a perfect cube of 2.

Note that the second and third properties will NEVER both be true for the same cube.

Answer 2

A cube has all of its edges equal to each other. Let x = length of one of the edges of a cube.
a) Volume of a cube = (x)(x)(x) = x^3 This is a perfect cube

b) Total surface aread = 6(x^2) This is NOT a perfect cube!

c) Total length of the edge = 12x this is NOT a perfect cube!

d) Number of vertices of solid cube = 8 = 2^3 This is a perfect cube

As far as I can see, only the volume and the number of vertices of a solid cube are perfect cube.

teddy boy

Answer 3

If the length of one side is x then

the volume is x^3
(however, unless x is an integer, then x^3 will not be a perfect cube)

the total surface area is 6*x^2 (there are six identical faces, each with an area of x^2), not a cube (unless x = 6)

there are 10 edges, each of length x: total length = 10x (a perfect cube only if x = 100)

etc

Answer 4

You can easily prove this
if side is x, length is x
then volume is x^3
total suface area is 6x^2
because you're told its a cube it has 8 vertexes.
the total length of the edges is 12x (perimeter).

The trick here is that you're told its a cube, and has side x. Since you're told side x, you can assume all sides are x because its a CUBE.

Im not sure that you wrote the whole question. But thats my answer anyways.

Answer 5

r u sure u have written the complete question? i dont think so..