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# Tricky problem watch

1. I'm stuck between the relationship between Volume and the surface area.

Ie V is proportional to an inverse function. But I forgot the inverse function. Any help?
2. Surely volume increases with surface area?

Geometrists and topologists come help out! Wait, are topologists interested in this stuff...
3. Volume increses with surface area. But the relationships is like double the length = 4x the surface area = 8x the volume.

And my brain is tired. and I think it is an inverse relationship
4. It will depend on the shape.
Interestingly, the surface area of a sphere is equal to the derivative of the volume. But i dont think its true generally.
5. So what about a general formula like x is proportional to xyz is that out of a question for the general shape?

What about the simple cube? Whats the general formula for that?
6. (Original post by 2776)
So what about a general formula like x is proportional to xyz is that out of a question for the general shape?

What about the simple cube? Whats the general formula for that?
Well you could add to the surface area without increasing the volume.
7. (Original post by 2776)
So what about a general formula like x is proportional to xyz is that out of a question for the general shape?

What about the simple cube? Whats the general formula for that?
Well, the volume increases by the linear scale factor cubed and the surface area by 6 * (linear scale factor squared), so if the volume is multiplied by some constant, n, then the linear scale factor is multiplied by n^(1/3) and, hence, the surfce area is multiplied by 6n^(2/3). That could be utter rubbish - check it first!

Ben
8. (Original post by 2776)
So what about a general formula like x is proportional to xyz is that out of a question for the general shape?

What about the simple cube? Whats the general formula for that?
If you deal with specific regular shapes, then yes, you can find a relationship. If the shape is irregular, there isnt a relationship. Consider a cube, but one face has a triangular indentation. By making that indentation larger, the surface area increases, but the volume decreases, but increasing the other lengths increases both volume and surface area. So there isnt a general rule.

For a cube, V = sqrt(SurfaceArea/6)^3
9. (Original post by JamesF)
If you deal with specific regular shapes, then yes, you can find a relationship. If the shape is irregular, there isnt a relationship. Consider a cube, but one face has a triangular indentation. By making that indentation larger, the surface area increases, but the volume decreases, but increasing the other lengths increases both volume and surface area. So there isnt a general rule.

For a cube, V = sqrt(SurfaceArea/6)^3
Good, I haven't made a fool of myself!

Ben
10. (Original post by Ben.S.)
Well, the volume increases by the linear scale factor cubed and the surface area by 6 * (linear scale factor squared), so if the volume is multiplied by some constant, n, then the linear scale factor is multiplied by n^(1/3) and, hence, the surfce area is multiplied by 6n^(2/3). That could be utter rubbish - check it first!

Ben
Yea, I think thats right.
11. (Original post by 2776)
Volume increses with surface area. But the relationships is like double the length = 4x the surface area = 8x the volume.

And my brain is tired. and I think it is an inverse relationship
do you mean (volume scale factor) = (length scale factor)³ = (area scale factor)^3/2?
12. (Original post by JamesF)
It will depend on the shape.
Interestingly, the surface area of a sphere is equal to the derivative of the volume. But i dont think its true generally.
is that a direct consequence of the divergence theorem?

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